Real projective orbifolds with ends and their deformation theory (draft. June, 2018) Suhyoung Choi

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1 Real projective orbifolds with ends and their deformation theory (draft. June, 2018) Suhyoung Choi DEPARTMENT OF MATHEMATICAL SCIENCES, KAIST, DAEJEON, SOUTH KOREA address:

2 2010 Mathematics Subject Classification. Primary 57M50; Secondary 53A20, 53C15 Key words and phrases. convex real projective structure, orbifold, character variety, deformations, deformation space, moduli space, Hilbert geometry, Finsler metric, group representation This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No ). ABSTRACT. In this monograph, we study the deformation spaces of convex real projective structures on orbifolds with ends. We need to put some restrictions on the end structure. We obtain an Ehresmann-Thurston-Weil type identification of the deformation spaces with appropriate end conditions with the character varieties with corresponding conditions.

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4 Contents Preface Acknowledgments Acronyms xi xv xvii Part 1. Introduction to orbifolds and real projective structures. 1 Chapter 1. Introduction Introduction A preview of the main results End structures Deformation spaces and the spaces of holonomy homomorphisms The Ehresmann-Thurston-Weil principle 22 Chapter 2. Preliminaries Preliminary definitions Affine orbifolds The needed linear algebra Convexity and convex domains The Vinberg duality of real projective orbifolds 57 Chapter 3. Examples of properly convex real projective orbifolds with ends: cusp openings History of examples Examples and computations The work of other groups Nicest cases Two specific examples 75 Part 2. The classification of radial and totally geodesic ends. 79 Chapter 4. Introduction to the theory of convex radial ends: classifying complete affine ends R-Ends Characterization of complete R-ends 85 Chapter 5. The affine action on properly convex domain with boundary in a totally geodesic hyperspace Affine actions The proximal flow. 97 vii

5 viii CONTENTS 5.3. Generalization Lens type T-ends 114 Chapter 6. Properly convex radial ends and totally geodesic ends: lens properties Main results The end theory The characterization of lens-shaped representations Convex cocompact actions of the p-end holonomy groups The uniform middle-eigenvalue conditions and the lens-shaped ends The properties of lens-shaped ends Duality and lens-shaped T-ends 143 Chapter 7. Application: The openness of the lens properties, and expansion and shrinking of end neighborhoods The openness of lens properties The end and the limit sets The strong irreducibility of the real projective orbifolds. 160 Chapter 8. The convex but nonproperly convex and non-complete-affine radial ends Introduction The transverse weak middle eigenvalue conditions for NPNC ends The general theory The non-discrete case The dual of NPNC-ends A result needed above 218 Part 3. The deformation space of convex real projective structures 221 Chapter 9. The openness of deformations Preliminary The local homeomorphism theorems Relationship to the deformation spaces in our earlier papers 246 Chapter 10. Relative hyperbolicity and strict convexity Some constructions associated with ends The strict SPC-structures and relative hyperbolicity Bowditch s method A topological result 264 Chapter 11. Openness and closedness Introduction The openness of the convex structures The proof of the openness The closedness of convex real projective structures General cases without the uniqueness condition: The proof of Theorem Appendix A. Projective abelian group actions on convex domains 291 A.1. Convex real projective orbifolds 291 A.2. The justification for weak middle eigenvalue conditions 300

6 CONTENTS ix Appendix. Index of Notations 303 Bibliography 305 Index 311

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8 Preface Let G be a Lie group acting transitively on a manifold X. An (X,G)-geometry is given by this pair. Furthermore, an (X,G)-structure on an orbifold or a manifold is an atlas of charts to X with transition maps in G. Here, we are concerned with G = PGL(n + 1,R) and X = RP n. Subjects of (X, G)-structures were popularized by Thurston and Goldman. These structures provide a way to understand representations and their deformations giving us viewpoints other than algebraic ones. Our deformation spaces often parameterize significant parts of the space of representations. Classically, conformally flat structures were studied much by differential geometers. Projectively flat structures were also studied from Cartan s time. However, our techniques are much different from their approaches. Convex real projective orbifolds are quotient spaces of convex domains on a projective space RP n by a discrete group projective automorphisms. Hyperbolic manifolds and many symmetric manifolds are natural examples. These can be deformed to one not coming from simple constructions. The study was initiated by Kuiper [103], Koszul [100], Benzécri [20], Vey [134], and Vinberg [136], accumulating some class of results. Closed manifolds or orbifolds admit many such structures as shown first by Kac-Vinberg [91], Goldman [76], and Cooper-Long-Thislethwaites [57], [58]. The some part of the theory for closed orbifolds were completed by Benoist [15] in 1990s. The topic of convex real projective structures on manifolds and orbifolds are currently developing. We present some parts of which we are working on. This book is mainly written for the researchers in this field. There are surprisingly many such structures coming from hyperbolic ones and deforming as shown by Vinberg for Coxeter orbifolds, Goldman for surfaces, and later by Cooper-Long-Thislethwaites for 3-manifolds. We compare these theories to the Mostow or Margulis type rigidity for symmetric spaces. The rigidity can be replaced by what is called the Ehresmann-Thurston-Weil principle that a union of components of the G-character varieties of the fundamental group of a manifold or orbifold M classifies the (X,G)-structures on M under the map where we define the deformation space hol : Def c (M) Hom c (π 1 (M),G)/G Def c (M) := {(x,g)-structures on M satisfying some conditions denoted by c}/ where is the isotopy equivalence relation, and Hom c (π 1 (M),G)/G is the subspace of the character variety Hom(π 1 (M),G)/G satisfying the corresponding conditions to c. For closed real projective orbifolds, it is widely thought that Benoist s work is quite an encompassing one. Hence, we won t say much about this topic. (See Choi-Lee-Marquis for a survey [53].) xi

9 xii PREFACE We focus on convex real projective orbifolds with ends, which we have now accumulated some number of examples. Basically, we will prove an Ehresmann-Thurston-Weil principle: We will show that the deformation space of properly convex real projective structures on an orbifold with some end conditions identifies under a map with the union of components of the subset of character spaces of the orbifold satisfying the corresponding conditions on ends holonomy groups. Our condition on the ends are probably very generic ones and we have many examples of such deformations. The book is divided into three parts: (I): We will give some introduction and survey our main results and give examples where our theory is applicable. (II): We will classify the types of ends we will work with. We use the middle eigenvalue condition. The condition is used to prove the preservation of the convexity of the deformations. (III): We will try to prove the Ehresmann-Thurston-Weil principle for our deformation spaces. First, we show the local homeomorphism property and then prove that the image is closed. We give an outline at the beginning of each part. We will use results of the whole of the monograph in Chapters 1 and 3. Also, we need the results of Chapter 8 in Part 2 in the later part of Chapter 4 in Part 2. Except for these the logical dependence of the monograph are as ordered by the monograph. Appendix A depends only on Chapter 2, and the results are used in the monograph except for Chapter 2. We mention that the definitions in Chapter 1 will be restated at most once more in various corresponding chapters to clarify with more materials. Chapter 2 will have many preliminary definitions which will be not repeated again. All definitions except for ones in Chapter 1 will not be repeated. The proofs in the book may be done by considering objects to be in S n and using the projective automorphism group SL ± (n + 1,R) or considering the objects to be in RP n and and using the projective automorphism group PGL(n + 1,R). We may add some remark after it when the proof is not completely obvious. We will use proof symbols: [SS n : ] at the end of the proof indicates that it is sufficient to prove for S n. [S n T: ] indicates that the version of the theorem, proposition, or lemma for S n implies one for RP n. [S n P: ] indicates that the proof of the theorem for S n implies one for RP n. For these proofs, we need to lift the objects to S n using Section If we don t need to go to S n to prove the result, we leave no mark except for the end of the proof. This book generalizes and simplifies the earlier preprints of the author. We were able to drop many conditions in the earlier versions of the theorems overcoming many limitations. Some of the results were announced in some survey articles [30] and [47]. Since the examples are easier to construct, even now, we will be studying orbifolds, a natural generalization of manifolds. Also, computations can be done fairly well for simple examples. We began our study with Coxeter orbifolds where the computations are probably the simplest possible. This is why we study orbifolds instead of just manifolds. As a motivation for our study, we say about some long term goal: Deforming a real projective structure on an orbifold to an unbounded situation results in the actions of the fundamental group on affine buildings which hopefully will lead us to some understanding of orbifolds and manifolds in particular of dimension three as indicated by Ballas, Cooper, Danciger, G. Lee, Leitner, Long, Thistlethwaite, and Tillmann.

10 PREFACE xiii There is a concurrent work by the group consisting of Cooper, Long, and Tillmann with Ballas and Leitner on the same subjects but with different conditions on ends. They impose the condition that the end fundamental group to be amenable. However, we do not require some conditions in this paper but instead we will use some type of eigenvalue conditions. Also, Crampon and Marquis as well as the above group investigated the convex real projective orbifolds with ends and with finite volume. However, we note that their deformation spaces are somewhat differently defined. We of course benefited much from their work and insight in this book and are very grateful for their generous help and guidance. June 2018 Daejeon, Suhyoung Choi

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12 Acknowledgments The study was begun with a conversation with Tillmann at Manifolds at Melbourne 2006 at University of Melbourne and I began to work on this seriously from my sabbatical at University of Melbourne from We thank Craig Hodgson and Gye-Seon Lee for working with me with many examples and their insights. This resulted in computing many examples and many valuable ideas. The idea of R-ends comes from the cooperation with them. We thank David Fried for helping me understand the issues with the distanced nature of the tubular actions and duality in Chapter 6. There is a subcase which David Fried proved but it is not included here. We thank Yves Carrière with the general approach to study the non-discrete cases for nonproperly convex ends in Chapter 8. The Lie group theoretical approach of Riemannian foliations was a key idea for the non-discrete quasijoined ends in Chapter 8. We thank Yves Benoist with some initial discussions on this topic, which were very helpful for Chapter 8 on quasi-joined actions and thank Bill Goldman and François Labourie for discussions resulting in Chapter 5 on affine actions on convex domains. We thank Daryl Cooper and Stephan Tillmann for explaining their work and help and we also thank Mickaël Crampon and Ludovic Marquis also. These discussions clarified some technical issues with ends. Their works obviously were influential here. We appreciate the hospitality of University of Melbourne during my sabbatical in the fall of 2008 where I started this research, and also the MSRI during the spring semester of 2015 where we first had the idea for writing a monograph. We also thank a few anonymous referees for pointing out some mistakes in our earlier preprints. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No ). xv

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14 Acronyms NPNC: nonproperly convex but not complete affine CA: complete affine PC: properly convex SPC: stable properly convex strict SPC: stable properly and strictly convex R-end: radial end T-end: totally geodesic end p-end: pseudo-end p-r-end: radial pseudo-end p-t-end: totally geodesic pseudo-end R-end: The end with type R assigned T -end: The end with type T assigned NA: non-annuluar property for ends IE: infinite-index end-fundamental-group condition [SS n ]: A proof symbol: Sufficient to prove for S n -version. [S n T]: A proof symbol: The S n -version of the proposition implies the RP n -version of the proposition. [S n P]: A proof symbol: The S n -version of the proof implies the RP n -version of the proof. xvii

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16 Part 1 Introduction to orbifolds and real projective structures.

17 The purpose of Part I is to survey the main results of the monograph and give some preliminary definitions. In Chapter 1, we give the introduction and state the main results of the monograph. Basic definitions associated convex real projective structures on orbifolds and some brief history will be given here. Then we state our mini-main result Corollary 1.3 illustrating the main focus of the Ehresmann-Thurston-Weil principle. Then we will basically illustrate the main results of the monograph. We will define ends and radial ends and discuss the relevant parts of the character varieties. Then we define stable properly convex real projective structures (SPC-structures) on orbifolds and strictly SPC-structures as well. We will prove the principle by dividing into two parts. Initially, we will show that the map hol is a local homeomorphism. Then we show that the image is closed. Finally, we end with stating state the results that are probably the most clear and simplistic in the monograph. In Chapter 2, we go over basic preliminary materials. First, we discuss the linear algebra and estimations using it. We discuss ends of orbifolds, convexity, the Benoist theory on convex divisible actions, and so on. Affine orbifolds and affine suspensions of real projective orbifolds are given. In Section 2.5 we discuss the dual orbifolds of a given convex real projective orbifold as given by Vinberg. In Chapter 3, we give examples where our theory applies to. Coxeter orbifolds and the orderbility theory for Coxeter orbifolds will be given. We discuss the work jointly done with Gye-Seon Lee, Hodgson and Greene. We state the work of Heusner-Porti on projective deformations of hyperbolic link complement. We also prove the nicest cases where nice results exist. These are the Coxeter orbifolds admitting complete hyperbolic structures.

18 CHAPTER 1 Introduction We define the real projective structures on orbifolds first describing as an immersion from the universal cover to RP n equivariant with respect to holonomy homomorphisms. We briefly discuss some history of the subject and the one for the orbifolds with ends. In Section 1.2, we will state a main result for a simplistic case. Here, every deformation space has some uniqueness property at each of the ends. To explain our main results better, we will now provide some background. In Section 1.3, we explain the ends, end neighborhoods, pseudo-ends, and pseudo-end fundamental groups. Then we explain the totally geodesic ends and lens associated with these. We explain the radial ends and lens-shaped ones and horospherical ones, which generalize the notions of cusps. These will be main types of ends we will consider. Then we will explain the middle eigenvalue conditions relating these ends with eigenvalue conditions. In Section 1.4, we will introduce the deformation spaces and character varieties. The character space will be refined by applying many end conditions corresponding to conditions for lens-shaped ends and horospherical ends. These will be shown to be semi-algebraic subsets. In Section 1.5, we present the main results of the monograph: Ehresmann-Thurston-Weil principles. We define the stable properly-convex real projective structures on an orbifold and their deformation space. We will divide into two parts proving the openness and the closedness. We will also use the fixed-point sections Introduction An n-dimensional orbifolds are basically objects finitely covered by n-dimensional manifolds. Hence, it is of form M/G for a manifold M and a group G acting on M properly discontinuously but not necessarily freely. We may even assume that G is finite in this monograph. We are interested in real projective structures on orbifolds, which are principally non-manifold ones. (See Section for precise definitions.) The real projective structures can be considered as torsion-free projectively flat affine connections on orbifolds. Another way to view is to consider these as an immersion from the universal cover Σ of an orbifold Σ to RP n equivariant with respect to a homomorphism h : π 1 (Σ) PGL(n+1,R) for the fundamental group π 1 (Σ) of Σ. (In this book, we assume that n 2 for convenience. ) These orbifolds have ends. We will study the cases when the ends are of specific type since otherwise it is almost impossible to study. The types that we consider are radial ones, i.e., R-ends, where end neighborhoods are foliated by concurrent projective geodesics. A hypersurface is a codimension-one submanifold or suborbifold of a manifold or an orbifold. Another type ones are totally geodesic ones, or T-ends, when the closure of the end neighborhood can be compactified by an ideal totally geodesic hypersurface in some ambient real projective orbifold. Kuiper, Benzécri, Koszul, Vey, and Vinberg might be the first people to consider these objects seriously as they are related to proper action of affine groups on affine cones in R n. 3

19 4 1. INTRODUCTION FIGURE 1. The developing images of convex RP 2 -structures on 2- orbifolds deformed from hyperbolic ones: S 2 (3,3,5) FIGURE 2. The developing images of convex RP 2 -structures on 2- orbifolds deformed from hyperbolic ones: D 2 (2,7). We note here, of course, the older study of affine structures on manifolds with many major open questions Real projective structures on manifolds and orbifolds. In general, the theory of geometric structures on manifolds with ends is not studied very well. We should try to obtain more results here and find what the appropriate conditions are. This question seems to be also related to how to make sense of the topological structures of ends in many other geometric structures such as ones modeled on symmetric spaces and so on. (See for example [92], [93], and [82].) We devote Chapter 3 to examples. Given a vector space V, we let P(V) denote the space obtained by taking the quotient space of V {O} under the equivalence relation v w for v, w V {O} iff v = s w, for s R {0}. We let [ v] denote the equivalence class of v V {O}. For a subspace W of V, we denote by P(W) the image of W {O} under the quotient map, also said to be a subspace. Recall that the projective linear group PGL(n + 1,R) acts on RP n, i.e., P(R n+1 ), in a standard manner. Let O be a noncompact strongly tame n-orbifold where the orbifold boundary is not necessarily empty. A real projective orbifold is an orbifold with a geometric structure modeled on (RP n,pgl(n + 1,R)). (See Sections and 2.1.5, [45], and Chapter 6 of [46].)

20 1.1. INTRODUCTION 5 A real projective orbifold also has the notion of projective geodesics as given by local charts and has a universal cover O where a deck transformation group π 1 (O) acting on. The underlying space of O is homeomorphic to the quotient space O/π 1 (O). A real projective structure on O gives us a so-called development pair (dev,h) where dev : O RP n is an immersion, called the developing map, and h : π 1 (O) PGL(n + 1,R) is a homomorphism, called a holonomy homomorphism, satisfying dev γ = h(γ) dev for γ π 1 (O). The pair (dev,h) is determined only up to the action g(dev,h( )) = (g dev,gh( )g 1 ) for g PGL(n + 1,R) and any chart in the atlas extends to a developing map. (See Section 3.4 of [132].) Let R n+1 denote the dual of R n+1. Let RP n denote the dual projective space P(R n+1 ). PGL(n + 1,R) acts on RP n by taking the inverse of the dual transformation. Then h : π 1 (O) PGL(n + 1,R) has a dual representation h : π 1 (O) PGL(n + 1,R) sending elements of π 1 (O) to the inverse of the dual transformation of R n+1. A projective map f : O 1 O 2 from a real projective orbifold O 1 to O 2 is a map so that for each p O 1, there are charts φ 1 : U 1 RP n, p U 1, and φ 2 : U 2 RP n where f (p) U 2 where f (U 1 ) U 2 and φ 2 φ1 1 is a projective map. For an element g PGL(n + 1,R), we denote (1.1) g [ w] := [ĝ( w)] for [ w] RP n or := [(ĝ T ) 1 ( w)] for [ w] RP n where ĝ is any element of SL ± (n + 1,R) mapping to g and ĝ T the transpose of ĝ. The complement of a codimension-one subspace of RP n can be identified with an affine space R n. The group of affine transformations of R n are the restrictions to R n of the group of projective transformations of RP n fixing the subspace. We call the complement an affine subspace. It has a geodesic structure of a standard affine space. A convex domain in RP n is a convex subset of an affine subspace. A properly convex domain in RP n is a convex domain contained in a precompact subset of an affine subspace. A strictly convex domain in RP n is a properly convex domain that contains no segment in the boundary. (We will use this definition of convexity except for Section A slightly general definition will be used for S n as defined in Section ) A convex real projective orbifold is a real projective orbifold projectively diffeomorphic to the quotient Ω/Γ where Ω is a convex domain in an affine subspace of RP n and Γ is a discrete group of projective automorphisms of Ω acting properly. If an open orbifold has a convex real projective structure, it is covered by a convex domain Ω in RP n. Equivalently, this means that the image of the developing map dev( O) for the universal cover O of O is a convex domain for the developing map dev with associated holonomy homomorphism h. Here we may assume dev( O) = Ω, and O is projectively diffeomorphic to dev( O)/h(π 1 (O)). In our discussions, since dev is an embedding and so is h, O will be regarded as an open domain in RP n and π 1 (O) as a subgroup of PGL(n + 1,R) in such cases. A convex real projective orbifold is properly convex if Ω is a properly convex

21 6 1. INTRODUCTION domain. (We will often drop real projective from convex real projective orbifold or manifold or properly convex real projective orbifold or manifold) (See Section for more details here.) REMARK 1.1 (Matrix convention). Given a vector space V, we denote by S(V) the quotient space of (V {O})/ where v w iff v = s w for s > 0. We will represent each element of PGL(n + 1,R) by a matrix of determinant ±1; i.e., PGL(n+1,R) = SL ± (n+1,r)/ ±I. Recall the covering map S n = S(R n+1 ) RP n. For each g PGL(n +1,R), there is a unique lift in SL ± (n +1,R) preserving each component of the inverse image of dev( O) under S n RP n. We will use this representative. (See Section and Section also.) One simple class of examples are obtained as follows: Let B be the interior of the standard conic in an affine subspace A n RP n. The group of projective automorphisms of B equals PO(1,n) PGL(n+1,R), and is the group of isometries of B with the Hilbert metric; that is, the Klein model of the hyperbolic space. Given any complete hyperbolic manifold (resp. orbifold) M, M is of form B/Γ for a discrete group Γ PO(1, n). Thus, M admits a convex projective structure. M with the real projective structures obtained here is called a hyperbolic manifold (resp. orbifold) Real projective structures on closed orbifolds. We will briefly say about the results on convex real projective structures on closed orbifolds here and their deformation spaces. (For a more detailed survey, see Choi-Lee-Marquis [53].) After Kuiper [103], Benzécri [20] started to consider higher dimensional convex real projective manifolds. Let Ω/Γ be a closed real projective orbifold. He showed [20] that Ω has C 1,ε -boundary and if Ω has C 2 -boundary, then Ω is the interior of a conic. However, Benzécri could not find examples not coming from classical geometry as Benoist informed us on one occasion. Kac-Vinberg [91] first found examples of deformations. Higher-dimensional examples were found by Vinberg [136]. Goldman [76] classified convex real projective structures on surfaces and showed that the deformation space is homeomorphic to cells. After this, Benoist [15], [16], [17], [18], and [19] more or less completed the study of these orbifolds including their deformation spaces. Inkang Kim [96] simultaneously proved some of these results for closed manifolds admitting hyperbolic structures. Recently, it was discovered by D. Cooper, D. Long, and M. Thistlethwaite [57], [58] that plenty of closed hyperbolic 3-manifolds deform to convex real projective 3-manifolds. These are closed manifolds. REMARK 1.2. Some surveys of the deformation spaces of real projective structures on closed orbifolds are given in Choi [46] and Choi-Lee-Marquis [53] Real projective structures on orbifolds with ends. Earlier near 2008, S. Tillmann found an example of a 3-orbifold obtained from pasting sides of a single ideal hyperbolic tetrahedron admitting a complete hyperbolic structure with cusps and a oneparameter family of real projective structure deformed from the hyperbolic one. (Porti- Tillmann [123] improved this to a two-dimensional family.) Also, Marquis [115], [116] completed the end theory of 2-orbifolds. Craig Hodgson, Gye-Seon Lee, and I earlier found a few examples of deformations: 3-dimensional ideal hyperbolic Coxeter orbifolds without edges of order 3 has at least 6-dimensional deformation spaces in [50]. (See Example 3.10.)

22 1.2. A PREVIEW OF THE MAIN RESULTS 7 Crampon and Marquis [61] and Cooper, Long, and Tillmann [60] have done similar study with the finite Hilbert volume condition. In this case, only possible ends are horospherical ones. The work here studies more general type ends while we have benefited from their work. We will see that there are examples where horospherical R-ends deform to lens-shaped R-ends and vice versa ( see also Example 3.10.) More recently, Cooper, Long, and Leitner [6] also have made progresses on the classification of the ends where they require that the ends have neighborhoods with nilpotent fundamental groups. (See [60], [108]. [107], [109], and [6].) In part 2, we will also classify the ends but with different conditions, some of which overlap with theirs. Their group is also producing the theory of the deformations with the above restrictions. See Cooper-Long-Tillmann [59], Ballas-Marquis [8], and Ballas-Cooper-Leitner [6]. However, we note that their definition of the deformation spaces are more complicated with added structures. But as Davis observed, there are many other types such as ones preserving subspaces of dimension greater than equal to 0. In fact, Cooper and Long found such an example from S/SL(3,Z) for the space S of unimodular positive definite bilinear forms. Since S is a properly convex domain in RP 5 and SL(3,Z) acts projectively, S/SL(3,Z) is a strongly tame properly convex real projective orbifold by the classical theory of lattices. These types of ends were compactified by Borel and Serre [24] for arithmetic manifolds. The ends are not of type studied here since these are not radial or totally geodesic ends. It remains how to see for which of these types of real orbifolds, nontrivial deformations exist or not. For example, we can consider examples such as complete hyperbolic manifolds and how to compute the deformation spaces. From Theorem 1 in [50] with Coxeter orbifolds, we know that a complete hyperbolic Coxeter orbifold always deforms nontrivially. There is a six dimensional space of deformations as convex real projective orbifolds with suitable ends. This will be explained in Chapter 3. Here a horospherical end changes into lens-shaped R-ends as first shown by Benoist [19]. (See also [45].) S. Ballas [4, 5] also produced some results along this line for hyperbolic 3-manifolds. Choi, Green, and Lee also found a class of examples of this types of deformations of convex real projective structures. We state this as Theorem (We will give more details on the examples in Chapter 3.) We conjecture that maybe these types of real projective orbifolds with R-ends might be very flexible. We have many examples to be explained in Chapter 3. There is also a class which we have not yet written down to be explain there also. Also, Ballas, Danciger, and Lee [7] announced that they have found much evidence for this very recently in Cooperfest at Berkeley in May Also, there are some related developments for the complex field C with the Ptolemy module in SnapPy SnapPy/ as developed by S. Garoufalidis, D. Thurston, and C. Zickert [72] A preview of the main results Our settings. Given an orbifold, we recall the notion of universal covering orbifold O with the orbifold covering map p O : O O and the deck transformation group π 1 (O) so that p O γ = p O for γ π 1 (O). (See [131], [26], [43] and [46].) We hope to generalize these theories to noncompact orbifolds with some particular conditions on ends. In fact, we are trying to generalize the class of complete hyperbolic manifolds with finite volumes. These are n-orbifolds with compact suborbifolds whose complements are diffeomorphic to intervals times closed (n 1)-dimensional orbifolds. Such orbifolds are said to be strongly tame orbifolds. An end neighborhood is a component of the complement of a compact subset not contained in any compact subset of the orbifold. An end E is an

23 8 1. INTRODUCTION equivalence class of compatible exiting sequences of end neighborhoods. Because of this, we can associate an (n 1)-orbifold at each end E and we define the end fundamental group π 1 (E) as a subgroup of the fundamental group π 1 (O) of the orbifold O which is the image of the fundamental group of a product end-neighborhood of E. (See Section for details.) We also put the condition on end neighborhoods being foliated by radial lines or to have totally geodesic ideal boundary. FIGURE 3. The singularities of the our orbifolds, the double of a tetrahedral reflection orbifold with orders 3, obtained by identifying two regular ideal tetrahedra by faces and Tillmann s orbifold in the 3-spheres, obtained from an ideal tetrahedra with four edges of angles π/6 and two edges of angles π/3 and gluing across the two edges by respective isometries. The white dots indicate the points removed. The edges are all of order 3. We concentrate on studying the ends that are well-behaved, i.e., ones that are foliated by lines or are totally geodesic. In this setting we wish to study the deformation spaces of the convex real projective structures on orbifolds with some boundary conditions using the character varieties. Our main aim is to identify the deformation space of convex real projective structures on an orbifold O with certain boundary conditions with open subset of a union of some semi-algebraic subsets of Hom(π 1 (O),PGL(n + 1,R))/PGL(n + 1,R) defined by conditions corresponding to the boundary conditions. Deformation spaces are defined in Section 1.4. This is an example of the so-called Ehresmann- Thurston-Weil principle [138]. The precise statements are given in Theorems 1.22 and See Definition 1.18 for the conditions (IE) and (NA). We use the notion of strict convexity with respect to ends as defined in Definition Our main result is the following corollary of Theorem 1.29: COROLLARY 1.3. Let O be a noncompact strongly tame SPC n-orbifold with generalized lens-shaped or horospherical ends and satisfies (IE) and (NA). Assume O = /0, and that the nilpotent normal subgroups of every finite-index subgroup of π 1 (O) are trivial. Then hol maps the deformation space CDef E,u,ce (O) of SPC-structures on O homeomorphically to a union of components of The same can be said for SDef E,u,ce (O). rep E,u,ce (π 1 (O),PGL(n + 1,R)).

24 1.3. END STRUCTURES 9 These terms will be defined more precisely later on in Sections and Roughly speaking, CDef E (O) (resp. SDef E (O)) is the deformation space of properly convex (resp. strictly convex) real projective structures with conditions on ends that each end holonomy group fixes a point. rep s E (π 1(O),PGL(n + 1,R)) is the space of characters each of whose end holonomy group fixes a point. CDef E,u,ce (O) (resp. SDef E,u,ce (O)) is the deformation space of properly convex (resp. strictly properly convex) real projective structures with conditions on ends that each end has a lens-cone neighborhood or a horospherical one, and each end holonomy group fixes a unique point. rep s E,u,ce (π 1(O),PGL(n + 1,R)) is the space of characters each of whose end holonomy group fixes a unique point and acts on a lens-cone or a horosphere. As explained in Chapter 3 devoted to examples, our main examples satisfy this condition: Suppose that a strongly tame properly convex 3-orbifold O with radial ends admits a finite volume complete hyperbolic structure and has radial ends only and any end fundamental group is generated by finite order elements. Since finite-volume hyperbolic n- orbifolds satisfy (IE) and (NA) (see P.151 of [113] for example), the theory simplifies by Corollary 3.25, i.e., each end is always generalized lens-shaped or horospherical, so that SDef E,u,ce (O) = SDef E (O). The space under hol maps homeomorphically to a union of components of rep E (π 1 (O),PGL(4,R)). For a strongly tame Coxeter orbifold O of dimension n 3 admitting a complete finite-volume hyperbolic structure hol is a homeomorphism from to a union of components of SDef E,u,ce (O) = SDef E (O) rep s E (π 1(O),PGL(n + 1,R)) by Corollary For this theory, we can consider a Coxeter orbifold based on a convex polytope admitting a complete hyperbolic structure with all edge orders equal to 3. More specifically, we can consider a hyperbolic ideal simplex or a hyperbolic ideal cube with such structures. (See Choi-Hodgson-Lee [50] for examples of 6-dimensional deformations.) The remarkable work of Cooper-Long-Tillman [59] concentrates on openness. Their ends have nilpotent holonomy, and the character spaces are augmented with the deformation spaces of ends End structures End fundamental groups. Recall that a strongly tame n-orbifold is one where the complement of a compact set is diffeomorphic to a union of (n 1)-dimensional orbifolds times intervals. Of course it can be compact.

25 10 1. INTRODUCTION Let O be a real projective orbifold with the universal cover O and the covering map po. Each end neighborhood U, diffeomorphic to S E (0,1) for an (n 1)-orbifold S E, of an end E lifts to a connected open set Ũ in O. A subgroup ΓŨ of Γ acts on Ũ where p 1 O (U) = g π 1 (O) g(ũ). Each component Ũ is said to be a proper pseudo-end neighborhood. An exiting sequence of sets U 1,U 2, in O is a sequence so that for each compact subset K of O there exists an integer N satisfying p 1 O (K) U i = /0 for i > N. A pseudo-end neighborhood sequence is an exiting sequence of proper pseudoend neighborhoods {U i i = 1,2,3,...}, where U i+1 U i for every i. Two pseudo-end sequences {U i } and {V j } are compatible if for each i, there exists J such that V j U i for every j, j > J and conversely for each j, there exists I such that U i V j for every i, i > I. A compatibility class of a proper pseudo-end sequence is called a pseudo-end of O. Each of these corresponds to an end of O under the universal covering map p O. For a pseudo-end Ẽ of O, we denote by ΓẼ the subgroup ΓŨ where U and Ũ is as above. We call ΓẼ a pseudo-end fundamental group. We will also denote it by π 1 (Ẽ). A pseudo-end neighborhood U of a pseudo-end Ẽ is a ΓẼ-invariant open set containing a proper pseudo-end neighborhood of Ẽ. A proper pseudo-end neighborhood is an example. (From now on, we will replace pseudo-end with the abbreviation p-end.) (See Section for details.) PROPOSITION 1.4. Let Ẽ be a pseudo-end of a strongly tame orbifold O. The p-end fundamental group ΓẼ of Ẽ is independent of the choice of U. PROOF. Given U and U that are end-neighborhoods for an end E, let Ũ and Ũ be p- end neighborhoods for a p-end Ẽ that are components of p 1 (U) and p 1 (U ) respectively. Let Ũ be the component of p 1 (U ) that is a p-end neighborhood of Ẽ. Then ΓŨ injects into ΓŨ since both are subgroups of Γ. Any G -path in U in the sense of Bridson-Haefliger [26] is homotopic to a G -path in U by a translation in the I-factor. Thus, π 1 (U ) π 1 (U) is surjective. Since Ũ is connected, any element γ of ΓŨ is represented by a G - path connecting x 0 to γ(x 0 ). (See Example 3.7 in Chapter III.G of [26].) Thus, ΓŨ is isomorphic to the image of π 1 (U) π 1 (O). Since ΓŨ is surjective to the image of of π 1 (U ) π 1 (O), it follows that ΓŨ is isomorphic to ΓŨ Totally geodesic ends. Suppose that an end E of a real projective orbifold O satisfies the following: The end has an end neighborhood homeomorphic to a closed connected (n 1)- dimensional orbifold B times a half-open interval (0,1]. The end neighborhood completes to a compact orbifold U diffeomorphic to B [0,1] in an ambient real projective orbifold. The subset of U corresponding to B {0} is the added boundary component.

26 1.3. END STRUCTURES 11 Each point of the added boundary component has a neighborhood projectively diffeomorphic to the quotient orbifold of an open set V in an affine half-space P so that V P /0 by a projective action of a finite group. This implies that the developing map extends to the universal cover of the orbifold with U attached. The completion is called a compactified end neighborhood of the end E. The boundary component S E is called the ideal boundary component of the end. Such ideal boundary components may not be uniquely determined as there are two projectively nonequivalent ways to add boundary components of elementary annuli (see Section 1.4 of [38]). Two compactified end neighborhoods of an end are equivalent if the end neighborhood contain a common end neighborhood whose compactification projectively embed into the compactified end neighborhoods. (See Definition 9.1 for more detail.) The equivalence class of compactified end-neighborhoods is called an totally geodesic end structure (T-end structure) for an end E. We also define as follows: The equivalence class of the chosen compactified end neighborhood is called an totally geodesic end-structure of the totally geodesic end. The choice of the end structure is equivalent to the choice of the ideal boundary component. We will also call the ideal boundary S E the end orbifold (or end ideal boundary component) of the end. RP n has a Riemannian metric of constant curvature called Fubini-Study metric. Recall that the universal cover O of O has a path-metric induced by dev : O RP n. We can Cauchy complete O of this path-metric. The Cauchy completion is called the Kuiper completion of O. (See [40].) Note we may sometimes use a lift dev : O S n lifting the developing map and use the same notation. A totally geodesic end (T-end) is an end equipped with a T-end structure. A totally geodesic p-end (T-p-end) is a p-end Ẽ corresponding to a T-end E. There is a totally geodesic (n 1)-dimensional domain SẼ in the Cauchy completion of O in the closure of a p-end neighborhood of Ẽ. Of course, SẼ covers S E. We call SẼ the p-end ideal boundary component. We will identify it with a domain in a hyperspace in RP n (resp. S n ) when dev is a fixed map to RP n (resp. S n ). DEFINITION 1.5. A lens is a properly convex domain L in RP n so that L is a union of two smooth strictly convex open disks. A properly convex domain L is a generalized lens if L is a union of two open disks one of which is strictly convex and smooth and the other is allowed to be just a topological disk. A lens-orbifold (or lens) is a compact quotient orbifold of a lens by a properly discontinuous action of a projective group Γ preserving each boundary component. Also, the domains or an orbifold projectively diffeomorphic to a lens or lens-orbifolds are lens. (Lens condition): The ideal boundary is realized as a totally geodesic suborbifold in the interior of a lens-orbifold in some ambient real projective orbifold of the cover of O corresponding to the end fundamental group. If the lens condition is satisfied for an T-end, we will call it the lens-shaped T-end. The intersection of a lens with O is called a lens end neighborhood of the T-end. A corresponding T-p-end is said to be lens-shaped T-p-end. In these cases, SẼ is a properly convex (n 1)-dimensional domain, and S E is a (n 1)-dimensional properly convex real projective orbifold. We will call the cover L of a lens orbifold containing S E the cocompactly acted lens (CA-lens) of SẼ where we assume

27 12 1. INTRODUCTION that π 1 (Ẽ) acts properly and cocompactly on the lens. L O is said to be lens p-end neighborhood of Ẽ or SẼ. We remark that for each component i L for i = 1,2 of L, i L/Γ is compact and both are homotopy equivalent up to a virtual manifold cover L/Γ of L/Γ for a finite index subgroup Γ. Also, the ideal boundary component of L/Γ has the same homotopy type as L/Γ and is a compact manifold. (See Selberg s Theorem 2.11.) Radial ends. A segment is a convex arc in a 1-dimensional subspace of RP n or S n. We will denote it by xy if x and y are endpoints. It is uniquely determined by x and y if x and y are not antipodal. In the following, all the sets are required to be inside an affine subspace A n and its closure to be either in RP n or S n. Let O denote the universal cover of O with the developing map dev. Suppose that an end E of a real projective orbifold satisfies the following: The end has an end neighborhood U foliated by properly embedded projective geodesics. Choose any map f : R [0,1] O so that f R {t} for each t is a geodesic leaf of such a foliation of U. Then f lifts to f : R [0,1] O where dev f R {t} for each t,t [0,1], maps to a geodesic in RP n ending at a point of concurrency common for every t. The foliation is called a radial foliation and leaves radial lines of E. Two such radial foliations F 1 and F 2 of radial end neighborhoods of an end are equivalent if the restrictions of F 1 and F 2 in an end neighborhood agree. A radial end structure is an equivalence class of radial foliations. A radial end is a end with a radial end structure. A radial p-end is a p-end with covering a radial end with induced foliation. Each lift of the radial foliation has a finite path-length induced from dev. A pseudo-end (p-end) vertex of a radial p-end neighborhood or a radial p-end is the common end of concurrent lift of leaves of the radial foliation which we obtain by Cauchy completion along the leaves. Note that dev always extend to the pseudo-end vertex. The p-end vertex is defined independently of the choice of dev. We will identify with a point of RP n (resp. S n ) if dev is a map to RP n (resp. S n ). A radial end structure for an end E gives us an end-compactification O E of O so that distinct rays end at distinct points. Here, O E has an obvious smooth structure. (See Definition 9.1 for more detail.) Let RP n 1 x denote the space of concurrent lines to a point x where RP n 1 x is projectively diffeomorphic to RP n 1. The real projective transformations fixing x induce real projective transformations of RP n 1 x. LEMMA 1.6. Let Ω be a properly convex domain with x bdω with a group Γ of discrete group of projective automorphisms of Ω. We regard dev as the inclusion map. Let Γ x be the subgroup fixing x. Let Û be the space of directions of lines in Ω ending at x. Then the following hold: The space Û is a convex subset of an affine subspace of RP n 1 x. Γ x acts properly discontinuously on Û if and only if every line in Ω ending at x maps to a properly embedding lines in Ω/Γ x. Suppose that O = Ω/Γ is a noncompact strongly tame orbifold. x identifies by dev to a p-end vertex of a radial p-end neighborhood if and only if Γ x acts properly discontinuously on Û. PROOF. Since Ω has a Hilbert metric, which is Finsler, Γ acts properly on Ω. (See [97].) The fact Û is convex is clear since Ω is convex. Two last parts are straightforward. [SS n ].

28 1.3. END STRUCTURES 13 Let Ω be a properly convex domain in RP n so that O = Ω/Γ for a discrete subgroup Γ of automorphisms of Ω. The space of radial lines in an R-end lifts to a space R x (Ω) of lines in Ω ending at a point x of bdω. By above Γ x acts properly on R x (Ω). The quotient space R x (Ω)/Γ x has an (n 1)-orbifold structure by Lemma 1.6 and the properness of the radial lines. The end orbifold Σ E associated with an R-end is defined as the space of radial lines in O. It is clear that Σ E can be identified with R x (Ω)/Γ x. The space of radial lines in an R-end has the local structure of RP n 1 since we can lift a local neighborhood to O, and these radial lines lift to lines developing into concurrent lines. The end orbifold has a unique induced real projective structure of one dimension lower. An n-dimensional submanifold L of an affine patch A n is said to be a prehoroball if it is strictly convex, and the boundary L is diffeomorphic to R n 1 and bdl L is a single point. The boundary L is said to be a pre-horosphere. Recall that an n-dimensional subdomain L of A n is a lens if L is a convex domain and L is a disjoint union of two smoothly strictly convex embedded open (n 1)-cells + L and L. A cone is a bounded domain D in an affine patch with a point in the boundary, called an end vertex v so that every other point x D has an open segment vx o D. A cone D is a join {v} A for a subset A of D if D is a union of segments starting from v and ending at A. (See Definition 2.41.) The cone {p} L over a lens-shaped domain L in A n, p Cl(L), is a lens-cone if it is a convex domain and satisfies {p} L = {p} + L for one boundary component + L of L and every segment from p to + L meets the other boundary component L of L at a unique point. As a consequence, each line segment from p to + L is transversal to + L. L is called the lens of the lens-cone. (Here different lenses may give the identical lens-cone.) Also, {p} L {p} is a manifold with boundary + L. Each of two boundary components of L is called a top or bottom hypersurface depending on whether it is further away from p or not. The top component is denoted by + L and the bottom one by L. An n-dimensional subdomain L of A n is a generalized lens if L is a convex domain and L is a disjoint union of a strictly convex smoothly embedded open (n 1)-cell L and an embedded open (n 1)-cell + L, which is not necessarily smooth. A cone {p} L is said to be a generalized lens-cone if {p} L = {p} + L, p Cl(L) and every segment from p to + L meets L at a unique point. A lens-cone will of course be considered a generalized lens-cone. We again define the top hypersurface and the bottom one as above. They are denoted by + L and L respectively. + L can be non-smooth L is required to be smooth. A totally-geodesic submanifold is a convex domain in a subspace. A cone-over a totally-geodesic submanifold D is a union of all segments with one endpoint a point x not in the hyperspace and the other endpoint in D. We denote it by {x} D. We apply these to ends: DEFINITION 1.7.

29 14 1. INTRODUCTION Horospherical R-end: An R-p-end Ẽ of O is pre-horospherical if it has a prehoroball in O as a p-end neighborhood, or equivalently an open p-end neighborhood U in O so that bdu O = bdu {v} for a boundary fixed point v. Ẽ is pre-horospherical if it has a pre-horoball in O as a p-end neighborhood. We require that the radial foliation of Ẽ is the one where each leaf ends at v. Lens-shaped R-end: An R-p-end Ẽ is lens-shaped (resp. generalized-lens-shaped ), if it has a p-end neighborhood that is a lens-cone (resp. generalized lens-cone) projectively diffeomorphic to the interior of L {v} under dev where L is a lens (resp. generalized lens) and h(π 1 (Ẽ)) acts properly and cocompactly on L, and every leaf of the radial foliation of the p-end neighborhood ends corresponds to a radial segment ending at v. In this case, the image L is said to be a compactly acted lens (resp. cocompactly acted generalized lens) of such a p-end. We abbreviate the terminology to be CA-lens (resp. generalized CA-lens). A p-end end neighborhood of Ẽ is (generalized) lens-shaped if it is a (generalized) lens-cone p-end neighborhood of Ẽ. An R-end of O is lens-shaped (resp. totally geodesic cone-shaped, generalized lensshaped ) if the corresponding R-p-end is lens-shaped (resp. totally geodesic cone-shaped, generalized lens-shaped ). An end neighborhood of an end E is (generalized) lens-shaped if so is a corresponding p-end neighborhood Ẽ. An end neighborhood is lens-shaped if it is a lens-shaped R-end neighborhood or T- end neighborhood. A p-end neighborhood is lens-shaped if it is a lens-shaped R-p-end neighborhood or T-p-end neighborhood. Of course it is redundant to say that R-end or T-end satisfies the lens condition depending on its radial or totally geodesic end structure. DEFINITION 1.8. A real projective orbifold with radial or totally geodesic ends is a strongly tame orbifold with a real projective structure where each end is an R-end or a T-end with an end structure given for each. An end of a real projective orbifold is (resp. generalized ) lens-shaped or pre-horospherical if it is a (resp. generalized) lens-shaped or pre-horospherical R-end or if it is a lens-shaped T-end Cusp ends. A parabolic algebra p is an algebra in a semi-simple Lie algebra g whose complexification contains a maximal solvable subalgebra of g (p of [133]). A parabolic group P of a semi-simple Lie group G is the full normalizer of a parabolic subalgebra. We shall show that all pre-horospherical ends are horospherical in Chapter 4. An ellipsoid in RP n = P(R n+1 ) (resp. in S n = S(R n+1 )) is the projection C {O} of the null cone C := { x R n+1 B( x, x) = 0} for a nondegenerate symmetric bilinear form B : R n+1 R n+1 R. Ellipsoids are always equivalent by projective automorphisms of RP n. An ellipsoid ball is the closed contractible domain in an affine subspace A n of RP n (resp. S n ) bounded by an ellipsoid contained in A n. A horoball is an ellipsoid ball with a point p of the boundary removed. An ellipsoid with a point p on it removed is called a horosphere. The vertex of the horosphere or the horoball is defined as p. Let U be a horoball with a vertex p in the boundary of B. A real projective orbifold that is projectively diffeomorphic to an orbifold U/Γ p for a discrete subgroup Γ p PO(1,n) fixing a point p bdb is called a horoball orbifold. A cusp or horospherical end is an end with an end neighborhood that is such an orbifold. A cusp group is a subgroup of a

30 1.4. DEFORMATION SPACES AND THE SPACES OF HOLONOMY HOMOMORPHISMS 15 parabolic subgroup of an isomorphic copy of PO(1,n) in PGL(n+1,R) or in SO + (1,n) in SL ± (n + 1,R). A cusp group is unipotent cusp-group if it is unipotent as well. By Corollary 4.5, an end is pre-horospherical if and only if it is a cusp end. We will use the term interchangeably but not in Chapter 4 where we will prove this fact Examples. EXAMPLE 1.9. The interior of a finite-volume hyperbolic n-orbifold with rank n 1 horospherical ends and totally geodesic boundary forms an example of a noncompact strongly tame properly convex real projective orbifold with radial or totally geodesic ends. For horospherical ends, the end orbifolds have Euclidean structures. (Also, we could allow hyperideal ends by attaching radial ends. See Section ) EXAMPLE For examples, if the end orbifold of an R-end E is a 2-orbifold based on a sphere with three singularities of order 3, then a line of singularity is a leaf of a radial foliation. End orbifolds of Tillmann s orbifold [123] and the the double of a tetrahedral reflection orbifold are examples. A double orbifold of a cube with edges having orders 3 only has eight such end orbifolds. (See Proposition 4.6 of [30] and their deformations are computed in [50]. Also, see Ryan Greene [80] for the theory.) Middle eigenvalue conditions. This corresponds to Part 2. In Chapters 4, 5, 6, and 8, we will classify ends that might possibly arise in properly convex orbifolds. We don t have a complete classification but we will use this. (Other group of people assume the nilpotency of the end holonomy group and they have a classification [109].) We will show that the uniform middle eigenvalue condition given in Definitions 6.4 and 6.5 in Chapter 6 implies that the lens-condition holds for an R-end or a T-end. The work in Part 2 will show that the lens-conditions are stable ones; that is, a sufficiently small perturbation of the structures will keep the lens conditions for each ends. Other types of ends are not stable. However, we will use a subspace of character varieties so that our conditions will be open conditions. That is, we can transition between horospherical and lens-shaped ends within the character spaces that we define. These will be justified in Part Deformation spaces and the spaces of holonomy homomorphisms An isotopy i : O O is a self-diffeomorphism so that there exists a smooth orbifold map J : O [0,1] O, so that i t : O O given by i t (x) = J(x,t) are self-diffeomorphisms for t [0,1] and i = i 1,i 0 = I O. strongly. We will extend this notion Two real projective structures µ 0 and µ 1 on O with R-ends or T-ends with end structures are isotopic if there is an isotopy i on O so that i (µ 0 ) = µ 1 where i (µ 0 ) is the induced structure from µ 0 by i where we require for each t i t (µ 0 ) has a radial end structure for each radial end, i t sends the radial end foliation for µ 0 from an R-end neighborhood to the radial end foliation for real projective structure µ t = i t (µ 0 ) with corresponding R-end neighborhoods, and i t extends to the union of totally geodesic ideal boundary components of µ 0 to that of µ t as a diffeomorphism.

31 16 1. INTRODUCTION To discuss the deformation spaces, we introduce the following notions. The end will be either assigned an R-type or a T -type. An R-type end is required to be radial. A T -type end is required to a T-end of lens-type or be horospherical. In this monograph, a strongly tame orbifold will always have such an assignment, and finite-covering maps will always respect the types. We will fix this type for our orbifolds in consideration. We define Def E (O) as the deformation space of real projective structures on O with end structures; more precisely, this is the quotient space of the real projective structures on O satisfying the above conditions for ends of type R and T under the isotopy equivalence relations. We define the topology more precisely in Section (See [43], [27] and [75] for more details. ) Character spaces of relevance The semi-algebraic properties of rep s (π 1 (O),PGL(n+1,R)) and related spaces. Since O is strongly tame, the fundamental group π 1 (O) is finitely generated. Let {g 1,...,g m } be a set of generators of π 1 (O). As usual Hom(π 1 (O),G) for a Lie group G has an algebraic topology as a subspace of G m. This topology is given by the notion of algebraic convergence {h i } h if h i (g j ) h(g j ) G for each j, j = 1,...,m. A conjugacy class of a representation is called a character in this monograph. The PGL(n + 1,R)-character variety rep(π 1 (O),PGL(n + 1,R)) is the quotient space of the homomorphism space where PGL(n + 1,R) acts by conjugation Similarly, we define Hom(π 1 (O),PGL(n + 1,R)) h( ) gh( )g 1 for g PGL(n + 1,R). rep(π 1 (O),SL ± (n + 1,R)) := Hom(π 1 (O),SL ± (n + 1,R))/SL ± (n + 1,R) as the SL ± (n + 1,R)-character variety. A representation or a character is stable if the orbit of it or its representative is closed and the stabilizer is finite under the conjugation action in Hom(π 1 (O),PGL(n + 1,R)) (resp. Hom(π 1 (O),SL ± (n + 1,R))). By Theorem 1.1 of [90], a representation ρ is stable if and only if it is irreducible and no proper parabolic subgroup contains the image of ρ. The stability and the irreducibility are open conditions in the Zariski topology. Also, if the image of ρ is Zariski dense, then ρ is stable. PGL(n + 1,R) acts properly on the open set of stable representations in Hom(π 1 (O),PGL(n + 1,R)). Similarly, SL ± (n + 1,R) acts so on Hom(π 1 (O),SL ± (n + 1,R)). (See [90] for more details.) A representation of a group G into PGL(n + 1,R) or SL ± (n + 1,R) is strongly irreducible if the image of every finite index subgroup of G is irreducible. Actually, many of the orbifolds have strongly irreducible and stable holonomy homomorphisms by Theorem An eigen-1-form of a linear transformation γ is a linear functional α in R n+1 so that α γ = λ α for some λ R. We recall the lifting of Remark 1.1.

32 1.4. DEFORMATION SPACES AND THE SPACES OF HOLONOMY HOMOMORPHISMS 17 Hom E (π 1 (O),PGL(n + 1,R)) to be the subspace of representations h satisfying The vertex condition for R-ends: h π 1 (Ẽ) has a nonzero common eigenvector of positive eigenvalues for a lift of h(π 1 (Ẽ)) in SL ± (n + 1,R) for each R-type p-end fundamental group π 1 (Ẽ), and The lens-condition for T -ends: h π 1 (Ẽ) acts on a hyperspace P for each T - type p-end fundamental group π 1 (Ẽ) and acts discontinuously and cocompactly on a lens L, a properly convex domain with L o P = L P /0 or a horoball tangent to P. We denote by Hom s (π 1 (O),PGL(n + 1,R)) the subspace of stable and irreducible representations, and define to be Hom s E (π 1(O),PGL(n + 1,R)) Hom E (π 1 (O),PGL(n + 1,R)) Hom s (π 1 (O),PGL(n + 1,R)). We define Hom E,u (π 1 (O),PGL(n + 1,R)) to be the subspace of representations h where h π 1 (Ẽ) has a unique common eigenspace of dimension 1 in R n+1 with positive eigenvalues for its lift in SL ± (n + 1,R) for each p-end holonomy group h(π 1 (Ẽ)) of R-type and h π 1 (Ẽ) has a common null-space P of eigen-1-forms satisfying the following: π 1 (Ẽ) acts properly and cocompactly on a lens L with L P with nonempty interior in P or H {p} for a horosphere H tangent to P at p and is unique such one for each p-end holonomy group h(π 1 (Ẽ)) of the p-end of T -type. We define to be Hom s E,u (π 1(O),PGL(n + 1,R)) Hom s (π 1 (O),PGL(n + 1,R) Hom E,u (π 1 (O),PGL(n + 1,R)). REMARK The above condition for type T generalizes the principal boundary condition for real projective surfaces. Since each π 1 (Ẽ) is finitely generated and there is only finitely many conjugacy classes of π 1 (Ẽ), Hom E (π 1 (O),PGL(n + 1,R)) is a closed semi-algebraic subset. Define Hom E, f (π 1 (O),PGL(n + 1,R)) to be a subset of Hom E (π 1 (O),PGL(n + 1,R))

33 18 1. INTRODUCTION so that each p-end holonomy group of R-p-ends acts on a finite set of points and each p-end holonomy group of T-p-ends acts on a finite set of hyperspaces. This is a semi-algebraic set since we can use discriminants of characteristic polynomials of the holonomy matrices of the generators of the end fundamental groups. PROPOSITION Hom E,u (π 1 (O),PGL(n + 1,R)) is an open subset of a semi-algebraic subset. So is Hom s E,u (π 1(O),PGL(n + 1,R)). PROOF. By Lemma 1.13, the condition that each p-end holonomy group has the unique fixed point is an open condition: Since if there are more than two fixed points for a holonomy, we can build higher dimensional proper invariant subspaces using these. The lowest dimensional subspace is open by the lemma. Let Ẽ be a T-p-end. Let h Hom E, f (π 1 (O),PGL(n + 1,R)), and let G := h(π 1 (Ẽ)). Assume that G is not a cusp group. Let P be a hyperspace where G acts on. We say that if G acts properly on a lens L with L P with nonempty interior in P, then P satisfies a lens-property. Then the holonomy group G acts on a one-dimensional family of hyperspaces or acts on a finite set of hyperspaces by looking at the generators. We may assume without loss of generality that G is torsion-free by taking a finite-index subgroup by Theorem Let P be a G-invariant hyperspace so that P L o /G is a closed aspherical (n 1)-manifold. In the first case, a lens L intersects a G-invariant plane P so that L P /0 since L is smooth and strictly convex. We may assume without loss of generality that N := L P is an aspherical (n 1)-dimensional manifold with boundary by transversality. However, L/G cannot be homotopy equivalent to N/G since N/G has nonempty boundary and hence different n-dimensional homology. Hence, G acts on finitely many hyperspaces. Proposition 7.4 implies that the condition of the existence of the hyperspace P satisfying the lens-property is an open condition in Hom E, f (π 1 (O),PGL(n + 1,R)). Suppose that there is another hyperspace P with a lens L satisfying the above properties. Then P P is also G-invariant. Hence, by Proposition 2.49, we obtain that Cl(P L) is a join K {k} for a properly convex domain K in P P and a point k in P P. Similarly, exchanging the role of P and P, we obtain that there is a point k P P fixed by G. G acts on the one-dimensional subspace S G containing k and k. There are no other fixed point on it since otherwise S G is the set of fixed points and G acts on any hyperspace containing P P and a point on S G. This contradicts our assumption. Hence, only k and k are fixed points in S G and P P and {k,k } contain all the fixed points of G. Now, k is the unique fixed point outside P. The existence of lens for P tells us that k must be a fixed point outside the closure of the lens. By Theorem 6.37, the existence of a lens for P tells us that every g G, the maximum norm of eigenvalues of g associated with k and P P is greater than that of k. Now, we switch the role of P and P. We can take a central element g with the largest norm of eigenvalue at k by the last item of Proposition 2.49 and the middle eigenvalue condition from Theorem We obtain a contradiction by considering g with the largest norm of eigenvalue at k. This cannot happen by the above paragraph above. Hence, P satisfying the lens-condition is unique.

34 1.4. DEFORMATION SPACES AND THE SPACES OF HOLONOMY HOMOMORPHISMS 19 Suppose that G is a cusp group. Then there exists a unique hyperspace P containing the fixed point of G tangent to horospheres where G acts on. Therefore, Hom E,u (π 1 (O),PGL(n + 1,R)) is in an open subset of a union of semi-algebraic subsets of Hom E, f (π 1 (O),PGL(n + 1,R)). LEMMA Let V be a semi-algebraic subset of PGL(n + 1,R) m (resp. SL ± (n + 1,R) m.) For each (g 1,...,g m ) V, there is a function E : V (g 1,...,g m ) E i (g i ) R,i = 1,...,m where E(g i ) R n+1 is the corresponding invariant subspace of g i. We assume that Then is an upper semi-continuous function on V. m E(g i ) {0} on every point of V. i=1 m V (g 1,...,g m ) dim E(g i ) PROOF. Since the limit subspace of E(g) is contained in an eigenspace of g, this follows. [SS n ] (1.2) We define to be the set We denote by the subspace of i=1 rep E (π 1 (O),PGL(n + 1,R)) Hom E (π 1 (O),PGL(n + 1,R))/PGL(n + 1,R). rep s E (π 1(O),PGL(n + 1,R)) rep E (π 1 (O),PGL(n + 1,R)) of stable and irreducible characters. We define rep E,u (π 1 (O),PGL(n + 1,R)) to be We define Hom E,u (π 1 (O),PGL(n + 1,R))/PGL(n + 1,R). rep s E,u (π 1(O),PGL(n + 1,R)) := rep s (π 1 (O),PGL(n + 1,R)) rep E,u (π 1 (O),PGL(n + 1,R)).

35 20 1. INTRODUCTION Let ρ Hom E (π 1 (E),PGL(n + 1,R)) where E is an end. Define Hom E,par (π 1 (E),PGL(n + 1,R)) to be the subspace of representations where π 1 (E) goes into a cusp group, i.e., a parabolic subgroup in a conjugated copy of PO(n,1). By Lemma 1.14, is a closed semi-algebraic set. Hom E,par (π 1 (E),PGL(n + 1,R)) LEMMA Hom E,par (G,PGL(n + 1,R)) is a closed algebraic set. PROOF. Let P be a maximal parabolic subgroup of a conjugated copy of PO(n+1,R) that fixes a point x. Then Hom(G,P) is a closed semi-algebraic set. equals a union another closed semi-algebraic set. Hom E,par (G,PGL(n + 1,R)) g PGL(n+1,R) Let E be an end orbifold of O. Given we define the following sets: Let E be an end of type R. Let Let Hom(G,gPg 1 ), ρ Hom E (π 1 (E),PGL(n + 1,R)), Hom E,RL (π 1 (E),PGL(n + 1,R)) denote the space of representations h of π 1 (E) where h(π 1 (E)) acts on a lenscone {p} L for a lens L and p Cl(L) of a p-end Ẽ corresponding to E and acts properly and cocompactly on the lens L itself. Thus, it is an open subspace of the above semi-algebraic set Hom E (π 1 (E),PGL(n + 1,R)) by Proposition 7.4. Let E denote an end of type T. Let Hom E,TL (π 1 (E),PGL(n + 1,R)) denote the space of totally geodesic representations h of π 1 (E) satisfying the following condition: h(π 1 (E)) acts on an lens L and a hyperspace P where L o P /0 and L/h(π 1 (E)) is a compact orbifold with two strictly convex boundary components. Hom E,TL (π 1 (E),PGL(n + 1,R)) is again an open subset of the semi-algebraic set by Proposition 7.4. Hom E (π 1 (E),PGL(n + 1,R)) R E : Hom(π 1 (O),PGL(n + 1,R)) h h π 1 (E) Hom(π 1 (E),PGL(n + 1,R)) be the restriction map to the p-end holonomy group h(π 1 (E)) corresponding to the end E of O. A representative set of p-ends of O is the subset of p-ends where each end of O has a corresponding p-end and a unique chosen corresponding p-end. Let R O denote the

36 1.4. DEFORMATION SPACES AND THE SPACES OF HOLONOMY HOMOMORPHISMS 21 representative set of p-ends of O of type R, and let T O denote the representative set of p-ends of O of type T. We define a more symmetric space: Hom E,ce (π 1 (O),PGL(n + 1,R)) to be ( ( R 1 E Hom E,par (π 1 (E),PGL(n + 1,R)) Hom E,RL (π 1 (E),PGL(n + 1,R))) ) E R O ( ( R 1 E Hom E,par (π 1 (E),PGL(n + 1,R)) Hom E,TL (π 1 (E),PGL(n + 1,R))) ). E T O The quotient space of the space under the conjugation under PGL(n + 1,R) is denoted by We define to be rep E,ce (π 1 (O),PGL(n + 1,R)). Hom s E,ce (π 1(O),PGL(n + 1,R)) Hom s (π 1 (O),PGL(n + 1,R)) Hom E,ce (π 1 (O),PGL(n + 1,R)). Hence, this is an open subset of a union of semialgebraic subsets in X := Hom s E (π 1(O),PGL(n + 1,R)). We don t claim that the union is open in X. These definitions allow for changes between horospherical ends to lens-shaped radial ones and totally geodesic ones. The quotient space of this space under the conjugation under PGL(n + 1,R is denoted by rep s E,ce (π 1(O),PGL(n + 1,R)). Since rep s E,u (π 1(O),PGL(n + 1,R)) is the Hausdorff quotient of the above set with the conjugation PGL(n + 1,R)-action, this is an open subset of a union of semi-algebraic subsets by Proposition 1.1 of [90]. We define Hom s E,u,ce (π 1(O),PGL(n + 1,R)) to be the subset Hom s E,u (π 1(O),PGL(n + 1,R)) ( ( R 1 ( E Hom E,par π1 (E),PGL(n + 1,R) ) ( Hom E,RL π1 (E),PGL(n + 1,R) ))) E R O ( ( R 1 ( E Hom E,par π1 (E),PGL(n + 1,R) ) ( Hom E,TL π1 (E),PGL(n + 1,R) ))). E T O Similarly to the above, Proposition 1.12 implies that PROPOSITION rep s E,u,ce (π 1(O),PGL(n + 1,R)) is an open subset of a union of semi-algebraic sets in rep s E,f (π 1(O),PGL(n + 1,R)).

37 22 1. INTRODUCTION 1.5. The Ehresmann-Thurston-Weil principle Note that elements of Def E (O) have characters in rep E (π 1 (O),PGL(n + 1,R)). Denote by Def E,u (O) the subspace of Def E (O) of equivalence classes of real projective structures with characters in rep E,u (π 1 (O),PGL(n + 1,R)) and each lift of radial rays of the p-end neighborhoods of R-type ends at the point fixed by the corresponding end holonomy group and the hyperspace determined by the ideal boundary component of T -p-end or the hyperspace tangent to the horosphere of the p-end of T -type coincides with that determined by the end holonomy group. Also, we denote by Def s E (O) Def E (O) and Def s E,u (O) Def E,u(O) the subspaces of equivalence classes of real projective structures with stable and irreducible characters. For technical reasons, we will be assuming O = /0 in most cases. Here, we are not yet concerned with convexity of orbifolds. The following map hol, the so-called Ehresmann- Thurston map, is induced by sending (dev,h) to the conjugacy class of h as isotopies preserve h: THEOREM 1.16 (Theorem 9.2). Let O be a noncompact strongly tame real projective n-orbifold with lens-shaped or horospherical ends with end structures and given types R or T. Assume O = /0. Then the following map is a local homeomorphism : Also, we define hol : Def s E,u (O) reps E,u (π 1(O),PGL(n + 1,R)). rep s E (π 1(O),SL ± (n + 1,R)),rep s E,u (π 1(O),SL ± (n + 1,R)) similarly to Section 1.4 where the uniqueness of the p-end vertex for each p-end is only up to the antipodal map. By lifting (dev,h) by the method of Section 2.1.6, we obtain that is a local homeomorphism. hol : Def s E,u (O) reps E,u (π 1(O),SL ± (n + 1,R)) REMARK The restrictions of end types are necessary for this theorem to hold. (See Goldman [75], Canary-Epstein-Green [27], Bergeron-Gelander [22] and Choi [43] for many versions of results for closed manifolds and orbifolds.) SPC-structures and its properties. DEFINITION For a strongly tame orbifold O, (IE) O or π 1 (O) satisfies the infinite-index end fundamental group condition (IE) if [π 1 (O) : π 1 (E)] = for the end fundamental group π 1 (E) of each end E. (NA) O or π 1 (O) satisfies the nonannular property (NA) if π 1 (Ẽ 1 ) π 1 (Ẽ 2 ) is finite for two distinct p-ends Ẽ 1,Ẽ 2 of O. (NA) implies that π 1 (E) contains every element g π 1 (O) normalizing h for an infinite order h π 1 (E) for an end fundamental group π 1 (E) of an end E. These conditions are satisfied by complete hyperbolic manifolds with cusps. These are group theoretical properties with respect to the end groups.

38 1.5. THE EHRESMANN-THURSTON-WEIL PRINCIPLE 23 DEFINITION 1.19 (Definition 7.8). An SPC-structure or a stable properly-convex real projective structure on an n-orbifold is a real projective structure so that the orbifold is projectively diffeomorphic to a quotient orbifold of a properly convex domain in RP n by a discrete group of projective automorphisms that is stable and irreducible. DEFINITION 1.20 (Definition 7.9). Suppose that O has an SPC-structure. Let Ũ be the inverse image in O of the union U of some choice of a collection of disjoint end neighborhoods of O with compact Cl(U). If every straight arc and every non-c 1 -point in bdo are contained in the closure of a component of Ũ, then O is said to be strictly convex with respect to the collection of the ends. And O is also said to have a strict SPC-structure with respect to the collection of ends. Notice that the definition depends on the choice of U. However, we will show that if each component U is required to be lens-shaped or horospherical, then we show that the definition is independent of U in Corollary 7.7. We will prove the following in Section 7.3. THEOREM Let O be a noncompact strongly tame properly convex real projective manifold with generalized lens-shaped or horospherical and satisfies (IE) and (NA). Then any finite-index subgroup of the holonomy group is irreducible and is not contained in a proper parabolic subgroup of PGL(n + 1,R) (resp. SL ± (n + 1,R)). That is, the holonomy is stable Main theorems. We now state our main results: We define Def s E,ce (O) to be the subspace of Def E (O) consisting of real projective structures with generalized lens-shaped or horospherical ends and stable irreducible holonomy homomorphisms. We define CDef E,ce (O) to be the subspace of Def E (O) consisting of SPC-structures with generalized lens-shaped or horospherical ends. We define CDef E,u,ce (O) to be the subspace of Def E,u (O) consisting of SPCstructures with generalized lens-shaped or horospherical ends. We define SDef E,ce (O) to be the subspace of Def E,ce (O) consisting of strict SPC-structures with lens-shaped or horospherical ends. We define SDef E,u,ce (O) to be the subspace of Def E,u,ce (O) consisting of strict SPC-structures with lens-shaped or horospherical ends. We remark that these spaces are dual to the same type of the spaces but we switch the R-end with T -ends and vice versa by Proposition Also by Corollary 7.19, for strict SPC-orbifolds with generalized lens-shaped or horospherical ends have only lens-shaped or horospherical ends. The following theorems are to be regarded as examples of the so-called Ehresmann- Thurston-Weil principle. THEOREM Let O be a noncompact strongly tame n-orbifold with generalized lens-shaped or horospherical ends. Assume O = /0. Suppose that O satisfies (IE) and (NA). Then the subspace CDef E,u,ce (O) Def s E,u,ce (O) is open. Suppose further that every finite-index subgroup of π 1 (O) contains no nontrivial infinite nilpotent normal subgroup. Then hol maps CDef E,u,ce (O) homeomorphically to a union of components of rep E,u,ce (π 1 (O),PGL(n + 1,R)).

39 24 1. INTRODUCTION THEOREM Let O be a strict SPC noncompact strongly tame n-dimensional orbifold with lens-shaped or horospherical ends and satisfies (IE) and (NA). Assume O = /0. Then π 1 (O) is relatively hyperbolic with respect to its end fundamental groups. The subspace SDef E,u,ce (O) Def s E,u,ce(O) of strict SPC-structures with lensshaped or horospherical ends is open. Suppose further that every finite-index subgroup of π 1 (O) contains no nontrivial infinite nilpotent normal subgroup. Then hol maps the deformation space SDef E,u,ce (O) of strict SPC-structures on O with lens-shaped or horospherical ends homeomorphically to a union of components of rep E,u,ce (π 1 (O),PGL(n + 1,R)). Theorems 1.22 and 1.23 are proved by dividing into the openness result in Section and the closedness result in Section Openness. For openness of SDef E,ce (O), we will make use of: COROLLARY 1.24 (Corollary 10.17). Assume that O is a noncompact strongly tame SPC-orbifold with generalized lens-shaped or horospherical ends and satisfies (IE) and (NA). Let E 1,...,E k be the ends of O. Assume O = /0. Then π 1 (O) is a relatively hyperbolic group with respect to the end groups π 1 (E 1 ),...,π 1 (E k ) if and only if O is strictly SPC with respect to ends E 1,...,E k. THEOREM Let O be a noncompact strongly tame real projective n-orbifold and satisfies (IE) and (NA). Assume O = /0. In Def s E,u,ce (O), the subspace CDef E,u,ce(O) of SPC-structures with generalized lens-shaped or horospherical ends is open, and so is SDef E,u,ce (O). PROOF. Hom s E,u (π 1(O),PGL(n + 1,R) is an open subset of Hom E (π 1 (O),PGL(n + 1,R)) by Proposition On Hom s E,u (π 1(O),PGL(n + 1,R)) has a uniqueness section defined by Lemma Now, Theorem 1.27 proves the result. We are given a properly real projective orbifold O with ends E 1,...,E e1 of R-type and E e1 +1,...,E e1 +e 2 of T -type. Let us choose representative p-ends Ẽ 1,...,Ẽ e1 and Ẽ e1 +1,...,Ẽ e1 +e 2. Again, e 1 is the number of R-type ends, and e 2 the number of T -type ends of O. We define a subspace of Hom E,ce (π 1 (O),PGL(n + 1,R)) to be as in Section Let V be an open subset of a union of semi-algebraic subsets of Hom s E (π 1(O),PGL(n + 1,R)) invariant under the conjugation action of PGL(n + 1,R) so that the following hold: one can choose a continuous section s (1) V : V (RPn ) e 1 sending a holonomy homomorphism to a common fixed point of ΓẼi for i = 1,...,e 1 and s (1) V satisfies s (1) V (gh( )g 1 ) = g s (1) V (h( )) for g PGL(n + 1,R).

40 s (1) V 1.5. THE EHRESMANN-THURSTON-WEIL PRINCIPLE 25 is said to be a fixed-point section. If Ẽ i for every i = 1,...,e 1 has a p-end neighborhood with a radial foliation with leaves developing into rays ending at the fixed point of the i-th factor of s (1) V, we say that radial end structures are determined by s (1) V. Again we assume that for the open subset V of a union of semi-algebraic subsets of Hom s E,ce (π 1(O),PGL(n + 1,R)) invariant under the conjugation action by PGL(n + 1,R), the following hold: s (2) V one can choose a continuous section s (2) V : V (RPn ) e 2 sending a holonomy homomorphism to a common dual fixed point of ΓẼi for i = e 1 + 1,...,e 1 + e 2, s (2) V satisfies s(2) V (gh( )g 1 ) = (g ) 1 s (2) V (h( )) for g PGL(n + 1,R), and letting P V (Ẽ i ) denote the null space of the i-th value of s (2) V for i = e 1 +1,...,e 1 + e 2, ΓẼi acts on the hyperspace P V (Ẽ i ) when Ẽ i is a T-p-end and acts on a horosphere tangent to P V (Ẽ i ) when Ẽ i is a horospherical R-p-end. is said to be a dual fixed-point section. If each Ẽ i for every i = e 1 + 1,...,e 1 + e 2 has a p-end neighborhood with the ideal boundary component in the hyperspace determined by the i-th factor of s (2) V provided Ẽ i is a T-end, or has a p-end neighborhood containing a ΓẼ-invariant horosphere tangent to the hyperspace determined by the i-th factor of s (2) V provided Ẽ i is a horospherical end, we say that end structures for the totally geodesic end are determined by s (2) We define s V : V (RP n ) e 1 (RP n ) e 2 as s (1) V s(2) V LEMMA We can define section V. and call it a fixing section. s u : Hom E,u,ce (π 1 (O),PGL(n + 1,R)) (RP n ) e 1 (RP n ) e 2 by choosing for each holonomy and each p-end the unique fixed point and the unique hyperspace as the images. PROOF. s u is a continuous function since a sequence of fixed points or dual fixed points of end holonomy group is a fixed point or a dual fixed point of the limit end holonomy group. We call s u the uniqueness section. Let V and s V : V (RP n ) e 1 (RP n ) e 2 be as above. We define Def s E,s V,ce (O) to be the subspace of Def E,s V (O) of real projective structures with generalized lens-shaped or horospherical end structures determined by s V, and stable irreducible holonomy homomorphisms in V. We define CDef E,sV,ce(O) to be the subspace consisting of SPC-structures with generalized lens-shaped or horospherical end structures determined by s V and holonomy homomorphisms in V in Def s E,s V,ce (O). We define SDef E,sV,ce(O) to be the subspace of consisting of strict SPC-structures with lens-shaped or horospherical end structures determined by s V and holonomy homomorphisms in V in Def s E,s V,ce (O).

41 26 1. INTRODUCTION THEOREM Let O be a noncompact strongly tame real projective n-orbifold with generalized lens-shaped or horospherical ends and satisfies (IE) and (NA). Assume O = /0. Choose an open PGL(n +1, R)-conjugation invariant subset of a union of semialgebraic subsets of V Hom s E,ce (π 1(O),PGL(n + 1,R)), and a fixing section s V : V (RP n ) e 1 (RP n ) e 2. Then CDef E,sV,ce(O) is open in Def s E,s V,ce (O), and so is SDef E,s V,ce(O). This is proved in Theorem By Theorems 1.25 and 1.16, we obtain: COROLLARY Let O be a noncompact strongly tame real projective n-orbifold with generalized lens-shaped or horospherical ends and satisfies (IE) and (NA). Assume O = /0. Then hol : CDef E,u,ce (O) rep s E,u,ce (π 1(O),PGL(n + 1,R)) is a local homeomorphism. Furthermore, if O has a strict SPC-structure with lens-shaped or horospherical ends, then so is hol : SDef E,u,ce (O) rep s E,u,ce (π 1(O),PGL(n + 1,R)) The closedness of convex real projective structures. The results here will be proved in Chapter 11 in Part 3. We recall rep s E (π 1(O),PGL(n + 1,R)) the subspace of stable irreducible characters of rep E (π 1 (O),PGL(n + 1,R)) which is shown to be the open subset of a semi-algebraic subset in Section , and denote by rep s E,u,ce (π 1(O),PGL(n + 1,R)) the subspace of stable irreducible characters of rep E,u,ce (π 1 (O),PGL(n + 1,R)), an open subset of a union of semialgebraic sets. THEOREM Let O be a noncompact strongly tame SPC n-orbifold with generalized lens-shaped or horospherical ends and satisfies (IE) and (NA). Assume O = /0, and that the nilpotent normal subgroups of every finite-index subgroup of π 1 (O) are trivial. Then the following hold : The deformation space CDef E,u,ce (O) of SPC-structures on O with generalized admissible ends maps under hol homeomorphically to a union of components of rep E,u,ce (π 1 (O),PGL(n + 1,R)). The deformation space SDef E,u,ce (O) of strict SPC-structures on O with lensshaped or horospherical ends maps under hol homeomorphically to the union of components of rep E,u,ce (π 1 (O),PGL(n + 1,R)). PROOF. Hom s E,u (π 1(O),PGL(n + 1,R) is an open subset of Hom E (π 1 (O),PGL(n + 1, R) by Proposition Corollary proves this by the existence of the uniqueness section in Section The following is probably the most general result. THEOREM 1.30 (Theorem 11.4). Let O be a noncompact strongly tame SPC-orbifold with generalized lens-shaped or horospherical ends and satisfies (IE) and (NA). Assume O = /0. Then

42 1.5. THE EHRESMANN-THURSTON-WEIL PRINCIPLE 27 Suppose that every finite-index subgroup of π 1 (O) contains no nontrivial infinite nilpotent normal subgroup and O = /0. Then hol maps the deformation space CDef E,ce (O) of SPC-structures on O with generalized lens-shaped or horospherical ends homeomorphically to a union of components of rep E,ce (π 1 (O),PGL(n + 1,R)). Suppose that every finite-index subgroup of π 1 (O) contains no nontrivial infinite nilpotent normal subgroup and O = /0. Then hol maps the deformation space SDef E,ce (O) of strict SPC-structures on O with lens-shaped or horospherical ends homeomorphically to a union of components of rep E,ce (π 1 (O),PGL(n + 1,R)). For example, these apply to the projective deformations of hyperbolic manifolds with torus boundary as in [7] Remarks. We give some remarks on our results here: The theory here is by no means exhaustive final words. We have a somewhat complicated theory of ends which are given in Part 2 of this monograph. Our boundary condition is very restrictive in some sense. With this it could be said that the above theory is not so surprising. Ballas, Cooper, Leitner, Long, and Tillmann have different restrictions on ends and they are working with manifolds. The associated end neighborhoods have nilpotent holonomy groups. (See [60], [59], [108], [107], [109], and [7]). They are currently developing the theory of ends and the deformation theory based on this assumption. Of course, we expect to benefit and thrive from many interactions between the theories as it happens in multitudes of fields. Originally, we developed the theory for orbifolds as given in papers of Choi [45], Choi, Hodgson, and Lee [50], and [32]. However, the recent examples of Ballas [4], [5], and Ballas, Danciger, and Lee [7] can be covered using fixing sections. Also, differently from the above work, we can allow ends with hyperbolic holonomy groups. REMARK As suggested by Mike Davis, one can look at ends with end holonomy groups acting on properly convex domains in totally geodesic subspaces of codimension between 2 and n 1. While they are perfectly reasonable to occur, in particular for Coxeter type orbifolds, we shall avoid these types as they are not understandable yet and we will hopefully study these in other papers. We will only be thinking of ends with end holonomy groups acting on codimension n or codimension 1 subspaces. However, we think that the other types of the ends do not change the theory present here in an essential way. (These are related to generalized Dehn surgery. See our article [52].) REMARK The lens condition is very natural and is a stable condition. We think that all other types of interests are limit of the ends with lens conditions. (See the above mentioned work of Ballas, Cooper and Leitner [6].) Horospherical ends occur naturally from hyperbolic orbifolds and it a limit of lens-shaped ends.

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44 CHAPTER 2 Preliminaries We will go over the basic theory. In Section 2.1, we discuss the Hilbert metrics, ends of the orbifolds, end fundamental groups, real projective structures on orbifolds, spherical real projective structures and liftings. In Section 2.2, we discuss that affine structures and affine suspensions of real projective orbifolds. In Section 2.3, we discuss the linear algebra for estimation, semi-simplicity and higher convergence groups. In Section 2.4, we explain the Benoist theory [17], [16], [18], and [19]. In particular, the strict-join decomposition of properly convex orbifolds will be explained. In Section 2.5, we explain the duality theory of Vinberg. In this monograph, we will be using the smooth category: that is, we will be using smooth maps and smooth charts and so on Preliminary definitions The following notation is used in the monograph. If A is a domain of a subspace of RP n or S n, we denote by bda the topological boundary in the subspace. The closure Cl(A) of a subset A of RP n or S n is the topological closure in RP n or in S n. A o will denote the manifold or orbifold interior of A. Define A for a manifold or orbifold A to be the manifold or orbifold boundary. (See Section ) Distances used. We will be using the standard elliptic metric d on RP n (resp. in S n ) where the set of geodesics coincides with the set of projective geodesics up to parameterizations. Sometimes, these are called Fubini-Study metrics. DEFINITION 2.1. Given a set, we define N ε (A) := {x S n d(x,a) < ε} ( resp. N ε (A) := {x RP n d(x,a) < ε}.) Given two compact subsets K 1 and K 2 of S n (resp. RP n ), we define the Hausdorff distance d H (K 1,K 2 ) between K 1 and K 2 to be The simple distance d(k 1,K 2 ) is defined as inf{ε > 0 K 2 N ε (K 1 ),K 1 N ε (K 2 )}. inf{d(x,y) x K 1,K 2 }. Recall that every sequence of compact sets {K i } in S n (resp. RP n ) has a convergent subsequence. We say that {K i } geometrically converges to a compact set K if d H (K i,k) 0 as i. The limit A is characterized as follows if it exists: A := {a H a is a limit point of some sequence {a i a i A i }}. (See Proposition E.12 of [10] for a proof since Chabauty topology for compact spaces is the Hausdorff topology and Munkres [122] for detail.) 29

45 30 2. PRELIMINARIES PROPOSITION 2.2 (Benedetti-Petronio). A sequence {A i } of compact sets in S n (resp. RP n ) converges to A in the Hausdorff topology if and only if the both of the following hold: If x i j A i j and x i j x, where i j, then x A. If x A, then there exists x i A i such that x i x. PROOF. Since S n and RP n are compact, the Chabauty topology is same as the Hausdorff topology. Hence, this is Proposition E.12 of Benedetti-Petronio [10]. LEMMA 2.3. Let g i be a sequence of elements of PGL(n+1,R) (resp. SL ± (n+1,r)) converging to g in PGL(n+1,R) (resp. SL ± (n+1,r)). Let K i be a sequence of compact set and let K be another one. Then K i K if and only if g i (K i ) g (K). PROOF. We use the above point description of the geometric limit. [SS n ] A n-hemisphere H in S n supports a domain D if H contains D. H is called a supporting hemisphere. An oriented hyperspace S in S n supports a domain D if the closed hemisphere bounded by S contains D. S is called a supporting hyperspace. If a supporting hyperspace contains a boundary point x of D, then it is called sharply-supporting hyperspace at x. If the boundary of a supporting n-hemisphere is sharply supporting at x, then the hemisphere is called sharply-supporting hemisphere at x. Note that we do not define these notions for RP n. The following is probably well-known. LEMMA 2.4. Suppose that a sequence K i of compact convex domain (resp. properly convex domain) in S n (resp. RP n ) converging to a compact convex set (resp. properly convex set) K with nonempty interior. Then bdk i bdk. PROOF. We prove for S n. Suppose that a point p is in bdk. Let B ε (p) be an open ε-ball of p. Suppose B ε (p) K i = /0 for infinitely many i. Then p cannot be a limit point of K by Proposition 2.2. This is a contradiction. Thus, B ε (p) K i /0 for i > N for some N. Suppose that B ε (p) K i for infinitely many i. Then since p bdk, some point in B(p) K is a limit point of some sequence p i, p i K i. This contradicts the first item of Proposition 2.2. Hence, B ε (p) bdk i /0 for i > M for some M. Then p is a limit of a sequence p i, p i bdk i. Conversely, suppose that a sequence p i j, p i j bdk i j where i j as j converges to p. Then p K clearly. If p K o, then there is ε,ε > 0, with B ε (p) K. Now, K i j has a sharply supporting closed hemisphere H i j at p i j with K i j H i j. Since p i j p, we may choose a subsequence k j so that H k j H and d H (H k j,h ) < ε/4 for a hemisphere H. Let q B 3ε/4 (p) H so that d H (q,h ) > ε/4. Hence, B ε/4 (q) B ε (p) H k j for all j. Since K k j H k j, no sequence q k j K k j converges to q. However since K k j K and q K, this is a contradiction to Proposition 2.2. For the RP n -version, we lift K i to S n to properly convex domains K i. Now, we may also choose a subsequence so that K i geometrically converges to a choice of a lift K of K by Proposition 2.2. Since K is properly convex, K is in a bounded subset of an affine subspace of S n Then the result follows from the S n -version. THEOREM 2.5. Suppose that K i and K are (resp. properly) convex compact balls of the same dimension in S n (resp. RP n ). Suppose that K i K. It follows that (2.1) K i K.

46 2.1. PRELIMINARY DEFINITIONS 31 PROOF. Since K i and K are of the same dimension, we find g i SL ± (n + 1,R) so that g i ( K i ) = K and g i g for g SL ± (n + 1,R). Then g i (K i ) g (K). Then g i (K i ) g (K) by Lemma 2.4. Hence, K i K. [S n P] The Hilbert metric. Let Ω be a properly convex open domain. A line or a subspace of dimension-one in RP n has a two-dimensional homogeneous coordinate system. Let [o,s,q, p] denote the cross ratio of four points on a line as defined by ō q s q s p ō p where ō, p, q, s denote respectively the first coordinates of the homogeneous coordinates of o, p,q,s provided that the second coordinates equal 1. Define a metric for p,q Ω, d Ω (p,q) = log [o,s,q, p] where o and s are endpoints of the maximal segment in Ω containing p,q where o,q separates p,s. The metric is one given by a Finsler metric. (See [97].) Assume that K i K geometrically for a sequence of properly convex compact domains K i and a properly convex compact domain K. Suppose that two sequences of points {x i x i K o i } and {y i y i K o i } converge to x,y Ko respectively. Since the end points of a maximal segments always are in K i and K i K, Theorem 2.5 shows that (2.2) d K o i (x i,y i ) d K o(x,y). We omit the details of the elementary proof. LEMMA 2.6. Let U be a convex subset of a properly convex domain V in S n (resp. RP n ). Let U ε := {x V d V (x,u) ε} for ε > 0. Then U ε is properly convex. PROOF. Given u,v U ε, we find w,t Ω so that d V (u,w) < ε,d V (v,t) < ε. Then each point of uv is within ε of wt U in the d V -metric. By Lemma 1.8 of [60], this follows. [SS n ] Topology of orbifolds. An n-dimensional orbifold structure on a Hausdorff space X is given by maximal collection of charts (U,φ,G) satisfying the following conditions: U is an open subset of R n and φ : U X is a map and G is a finite group acting on U, the chart φ : U X induces a homeomorphism U/G to an open subset of X, the sets of form φ(u) covers X. for any pair of models (U,φ,G) and (V,ψ,H) with an inclusion map ι : φ(u) φ(v ) lifts to an embedding U V equivariant with respect to an injective homomorphism G H. (compatibility condition) An orbifold O is a topological space with an orbifold structure. The boundary O of an orbifold is defined as the set of points with only half-open sets as models. (These are often distinct from topological boundary.) Orbifolds are stratified by manifolds. Let O denote an n-dimensional orbifold with finitely many ends where end neighborhoods are homeomorphic to closed (n 1)-dimensional orbifolds times an open interval. We will require that O is strongly tame; that is, O has a

47 32 2. PRELIMINARIES compact suborbifold K so that O K is a disjoint union of end neighborhoods homeomorphic to closed (n 1)-dimensional orbifolds multiplied by open intervals. Hence O is a compact suborbifold. (See [131], [1], [94] and [46] for details.) An orbifold covering map p : O 1 O is a map so that for any point on O, there is a connected open set U X with model (Ũ,φ,G) as above whose inverse image p 1 (U) is a union of connected open set U i of O 1 with models (Ũ,φ i,g i ) for a subgroup G i G and the induced chart φ i : Ũ Ũ i. We say that an orbifold is a manifold if it has a subatlas of charts with trivial local groups. We will consider good orbifolds only, i.e., covered by simply connected manifolds. In this case, the universal covering orbifold O is a manifold with an orbifold covering map p O : O O. The group of deck transformations will be denote by π 1 (O) or Γ, and is said to be the fundamental group of O. They act properly discontinuously on O but not necessarily freely Geometric structures on orbifolds. An (X, G)-structure on an orbifold O is an atlas of charts from open subsets of X with finite subgroups of G acting on them, and the inclusions always lift to restrictions of elements of G in open subsets of X. This is equivalent to saying that the orbifold O has a simply connected manifold cover O with an immersion D : O X and the fundamental group π 1 (O) acts on O properly discontinuously so that h : π 1 (O) G is a homomorphism satisfying D γ = h(γ) D for each γ π 1 (O). Here, π 1 (O) is allowed to have fixed points. (We shall use this second definition here.) (D,h( )) is called a development pair and for a given (X,G)-structure, it is determined only up to an action (D,h( )) (k D,kh( )k 1 ) for k G. Conversely, a development pair completely determines the (X, G)-structure. (See Thurston [132] for the general theory of geometric structures.) Thurston showed that an orbifold with an (X, G)-structure is always good, i.e., covered by a manifold with an (X,G)-structure Real projective structures on orbifolds. A cone C in R n+1 {O} is a subset so that given a vector x C, sx C for every s R +. A convex cone is a cone that is a convex subset of R n+1 in the usual sense. A properly convex cone is a convex cone not containing a complete affine line. Recall the real projective space RP n is defined as R n+1 {O} under the quotient relation v w iff v = s w for s R {O}. Given a vector v R n+1 {O}, we denote by [ v] RP n the equivalence class. Let Π : R n+1 {O} RP n denote the projection. Given a connected subset A of an affine subspace of RP n, a cone C(A) R n+1 of A is given as a connected cone in R n+1 mapping onto A under the projection Π : R n+1 {O} RP n. C(A) is unique up to the antipodal map A : R n+1 R n+1 given by v v. The general linear group GL(n + 1,R) acts on R n+1 and PGL(n + 1,R) acts faithfully on RP n. Denote by R + = {r R r > 0}. The real projective sphere S n is defined as the quotient of R n+1 {O} under the quotient relation v w iff v = s w for s R +. We will also use S n as the double cover of RP n. Then Aut(S n ), isomorphic to the subgroup SL ± (n + 1,R) of GL(n + 1,R) of determinant ±1, double-covers PGL(n + 1,R). Aut(S n ) acts as a group of projective automorphisms of S n. A projective map of a real projective orbifold to another is a map that is projective by charts to RP n. Let Π : R n+1 {O} RP n

48 2.1. PRELIMINARY DEFINITIONS 33 be a projection and let Π : R n+1 {O} S n denote one for S n. An infinite subgroup Γ of PGL(n + 1,R) (resp. SL ± (n + 1,R)) is strongly irreducible if every finite-index subgroup is irreducible. A subspace S of RP n (resp. S n ) is the image of a subspace with the origin removed under the projection Π (resp. Π ). A line in RP n or S n is an embedded arc in a 1-dimensional subspace. A projective geodesic is an arc in a projective orbifold developing into a line in RP n or to a one-dimensional subspace of S n. A great segment is an embedded geodesic connecting a pair of antipodal points in S n or the complement of a point in a 1-dimensional subspace in RP n. Sometimes open great segment is called a complete affine line. An affine subspace A n can be identified with the complement of a codimension-one subspace RP n 1 so that the geodesic structures are same up to parameterizations. A convex subset of RP n is a convex subset of an affine subspace in this paper. A properly convex subset of RP n is a precompact convex subset of an affine subspace. R n identifies with an open half-space in S n defined by a linear function on R n+1. (In this paper an affine subspace is either embedded in RP n or S n.) An i-dimensional complete affine subspace is a subset of a projective orbifold projectively diffeomorphic to an i-dimensional affine subspace in some affine subspace A n of RP n or S n. Again an affine subspace in S n is a lift of an affine subspace in RP n, which is the interior of an n-hemisphere. Convexity and proper convexity in S n are defined in the same way as in RP n. The complement of a codimension-one subspace W in RP n can be considered an affine space A n by correspondence [1,x 1,...,x n ] (x 1,...,x n ) for a coordinate system where W is given by x 0 = 0. The group Aff(A n ) of projective automorphisms acting on A n is identical with the group of affine transformations of form x A x + b for a linear map A : R n R n and b R n. The projective geodesics and the affine geodesics agree up to parametrizations. A subset A of RP n or S n spans a subspace S if S is the smallest subspace containing A. We write S = A. We will consider an orbifold O with a real projective structure: This can be expressed as having a pair (dev,h) where dev : O RP n is an immersion equivariant with respect to the homomorphism h : π 1 (O) PGL(n + 1,R) where O is the universal cover and π 1 (O) is the group of deck transformations acting on O. (dev,h) is only determined up to an action of PGL(n + 1,R) given by g (dev,h( )) = (g dev,gh( )g 1 ) for g PGL(n + 1,R). dev is said to be a developing map and h is said to be a holonomy homomorphism and (dev,h) is called a development pair. We will usually use only one pair where dev is an embedding for this paper and hence identify O with its image. A holonomy is an image of an element under h. The holonomy group is the image group h(π 1 (O)). The Klein model of the hyperbolic geometry is given as follows: Let x 0,x 1,...,x n denote the standard coordinates of R n+1. Let B be the interior in RP n or S n of a standard ball that is the image of the positive cone of x0 2 > x x2 n in R n+1. Then B can be identified with a hyperbolic n-space. The group of isometries of the hyperbolic space equals

49 34 2. PRELIMINARIES the group Aut(B) of projective automorphisms acting on B. Thus, a complete hyperbolic manifold carries a unique real projective structure and is denoted by B/Γ for Γ Aut(B). Actually, g(b) for any g PGL(n + 1,R) will serve as a Klein model of the hyperbolic space, and Aut(gB) = gaut(b)g 1 is the isometry group. (See [46] for details.) An orbifold O is convex (resp. properly convex and complete affine) if O is a convex domain (resp. a properly convex domain and an affine subspace.). A totally geodesic hypersurface A in O (resp. O) is a subspace of codimension-one where each point p in A has a neighborhood U in O (resp. O) with a chart φ so that φ A has the image in a hyperspace Oriented real projective structures. DEFINITION 2.7. A convex segment is an arc contained in a great segment. We use a slightly different definition of convexity for S n. A convex subset of S n is a subset A where every pair of points of A connected by a convex segment. It is easy to see that a convex subset of S n is either contained in an affine subspace or is in a closed hemisphere or it is a great sphere of dimension 1. In the first case, the set embeds to a convex set in RP n under the covering map. PROPOSITION 2.8. A closed convex subset K of S n is either a great sphere S i 0 of dimension i 0 1, or is contained in a closed hemisphere H i 0 in S i 0 and is one of the following: There exists a great sphere S j 0 of dimension j 0 0 in the boundary K and a compact properly convex domain K K in the independent subspace of S j 0 and K = S j 0 K K, a strict join. K is a properly convex domain in the interior of i hemisphere. for some i. PROOF. Let S i 0 be the span of K. Then K o is not an empty domain in S i 0. d(x,k) is a function on S i 0. Suppose that S i 0 K /0. Choose a maximum point x 0. If the maximum is < π/2, then the elliptic geometry tells us that there at least two point y,z of K closest to x 0 of same distance from x 0 since otherwise we can increase the value of d(,k) by moving x 0 slightly. Then there is a closer point on xy o in K to x 0. This is a contradiction. Hence, K = S i 0. Otherwise, K is a subset of an i 0 -hemisphere in S i 0. (See [37] also.) The second part follows from Section 1.4 of [29]. (See also [63].) Hence, we obtain S j 0 in K and K is a union of j hemispheres with common boundary S j 0. By choosing an independent subspace S n j0 1 to S j 0, each j hemisphere in K is transversal to S n j0 1 and hence meets it in a unique point. We let K K denote the set of intersection points. Therefore, K = S j 0 K K. Recall that SL ± (n + 1,R) is isomorphic to GL(n + 1,R)/R +. Then this group acts on S n to be seen as a quotient space of R n+1 {O} by the equivalence relation v w, v, w R n+1 {O} if v = s w for s R +. We let [ v] denote the equivalence class of v R n+1 {O}. Given a vector subspace V R n+1, we denote by S(V ) the image of V {O} under the quotient map. The image is called a subspace. A set of antipodal points is a subspace of dimension 0. There is a double covering map S n RP n with the deck transformation group generated by A. This gives a projective structure on S n. The group of projective automorphisms is identified with SL ± (n + 1,R). The notion of geodesics are defined as in the projective geometry: they correspond to arcs in great circles in S n.

50 2.1. PRELIMINARY DEFINITIONS 35 A collection of subspaces S(V 1 ),...,S(V m ) (resp. P(V 1 ),...,P(V m )) are independent if the subspaces V 1,...,V m are independent. The group SL ± (n+1,r) of linear transformations of determinant ±1 maps to the projective group PGL(n + 1,R) by a double covering homomorphism ˆq, and SL ± (n + 1,R) acts on S n lifting the projective transformations. The elements are also projective transformations. We now discuss the standard lifting: Given a real projective structure on O, there is a development pair (dev,h) where dev : O RP n is an immersion and h : π 1 (O) PGL(n + 1,R) is a homomorphism. Since S n RP n is a covering map and O is a simply connected manifold, O being a good orbifold, there exists a lift dev : O S n unique up to the action of {I, I}. This induces an spherical real projective structure on O and dev is a developing map for this real projective structure. Given a deck transformation γ : O O, the composition dev γ is again a developing map for the real projective structure and hence equals h (γ) dev for h (γ) SL ± (n + 1,R). We verify that h : π 1 (O) SL ± (n + 1,R) is a homomorphism. Hence, (dev,h ) gives us a spherical real projective structure, which induces the original real projective structure. Given a projective structure where dev : O RP n is an embedding to a properly convex open subset D, the developing map dev lifts to an embedding dev : O S n to an open domain D without any pair of antipodal points. D is determined up to A. We will identify O with D or A (D) and π 1 (O) with Γ. Then Γ lifts to a subgroup Γ of SL ± (n+1,r) acting faithfully and discretely on O. There is a unique way to lift so that D/Γ is projectively diffeomorphic to O/Γ. THEOREM 2.9. There is a one-to-one correspondence between the space of real projective structures on an orbifold O with the space of oriented real projective structures on O. Moreover, a real projective diffeomorphism of real projective orbifolds is an (S n,sl ± (n+ 1,R))-diffeomorphism of oriented real projective orbifolds and vice versa. PROOF. Straightforward. See p. 143 of Thurston [132]. See Section also. Again, we can define the radial end structures and totally geodesic ideal boundary for spherical real projective structures and also horospherical end neighborhoods in obvious ways. They correspond in the direct way in the following results also. PROPOSITION 2.10 (Selberg-Malcev). The holonomy group of a convex real projective orbifold is residually finite. PROOF. In this case, dev : O RP n always lifts an embedding to a domain in S n. Γ also lifts to a projective automorphisms of the domain in SL ± (n + 1,R). The lifted group is residually finite by by Malcev [111]. Hence, Γ is thus always residually finite. [SS n ] THEOREM 2.11 (Selberg). A real projective orbifold S is covered finitely by a real projective manifold M and S is real projectively diffeomorphic to M/G 1 for a finite group G 1 of real projective automorphisms of M. An affine orbifold S is covered finitely by an affine manifold N, and S is affinely diffeomorphic to N/G 2 for a finite group G 2 of affine automorphisms of N. PROOF. Since Aff(R n ) is a subgroup of a general linear group, Selberg s Lemma [126] shows that there exists a torsion-free subgroup of the deck transformation group. We can choose the group to be a normal subgroup and the second item follows.

51 36 2. PRELIMINARIES A real projective structure induces an (S n,sl ± (n + 1,R))-structure and vice versa by Theorem 2.9. Also a real projective diffeomorphism of orbifolds is an (S n,sl ± (n + 1, R))-diffeomorphism and vice versa. We regard the real projective structures on S and M as (S n,sl ± (n + 1,R))-structures. We are done by Selberg s lemma [126] that a finitely generated subgroup of a general linear group has a torsion-free normal subgroup of finiteindex. [SS n ] A comment on lifting real projective structures and conventions. We sharpen Theorem 2.9. Let SL (n + 1,R) denote the component of SL ± (n + 1,R) not containing I. A projective automorphism g of S n is orientation preserving if and only if g has a matrix in SL(n + 1,R). For even n, the quotient map SL(n + 1,R) PGL(n + 1,R) is an isomorphism and so is the map SL (n + 1,R) PGL(n + 1,R) for the component of SL ± (n+1,r) with determinants equal to 1. For odd n, the quotient map SL(n+1,R) PGL(n + 1,R) is a 2 to 1 covering map onto its image component with deck transformations given by A ±A. THEOREM Let M be a strongly tame orbifold. Suppose that h : π 1 (M) PGL(n + 1,R) is a holonomy homomorphism of a real projective structure on M with radial or lens-shaped totally geodesic ends. Then the following hold : Suppose that M is orientable. We can lift to a homomorphism h : π 1 (M) SL(n + 1,R), which is a holonomy homomorphism of the (S n,sl ± (n + 1,R))- structure lifting the real projective structure. Suppose that M is not orientable. Then we can lift h to a homomorphism h : π 1 (M) SL ± (n+1,r) that is the holonomy homomorphism of the (S n,sl ± (n+ 1, R))-structure lifting the real projective structure so that the condition ( ) is satisfied. ( ) a deck transformation goes to a negative determinant matrix if and only if it reverses orientations. In general a lift h is unique if we require it to be the holonomy homomorphism of the lifted structure. For even n, the lifting is unique if we require the condition ( ). PROOF. Recall SL(n + 1, R) is the group of orientation-preserving linear automorphisms of R n+1 and hence is precisely the group of orientation-preserving projective automorphisms of S n. Since the deck transformations of the universal cover M of the lifted (S n,sl ± (n + 1,R))-orbifold are orientation-preserving, the holonomy of the lift are in SL(n + 1,R). We use as h the holonomy homomorphism of the lifted structure. For the second part, we can double cover M by an orientable orbifold M with an orientation-reversing Z 2 -action of the projective automorphism group generated by φ : M M. φ lifts to φ : M M for the universal covering manifold M = M and hence h( φ) dev = dev φ for the developing map dev and the holonomy h( φ) SL (n + 1,R). Then it follows from the first item since dev preserves orientations for a given orientation of M. (See p. 143 of Thurston [132].) The proof of uniqueness is straightforward. REMARK 2.13 (Convention on using spherical real projective structures). Suppose we are given a convex real projective orbifold of form Ω/Γ for Ω a convex domain in RP n and Γ a subgroup of PGL(n + 1,R). We can also think of Ω as a domain in S n and Γ SL ± (n + 1,R). We can think of them in both ways and we will use a convenient one for the purpose.

52 2.1. PRELIMINARY DEFINITIONS Convex hulls. DEFINITION Given a subset K of a convex domain Ω of an affine subspace A n in S n (resp. RP n ), the convex hull C H (K) of K is defined as the smallest convex set containing K in Cl(Ω) A n where we required Cl(Ω) A n. The convex hull is well-defined as long as Ω is properly convex. Otherwise, it may be not. Often when Ω is a properly convex domain, we will take the closure Cl(Ω) instead of Ω usually. This does not change the convex hull. (Usually it will be clear what Ω is by context but we will mention these.) For RP n, the convex hull depends on Ω but one can check that the convex hull is well-defined on S n as long as Ω is properly convex p-ends, p-end neighborhoods, and p-end groups. By strong tameness, O has only finitely many ends E 1,...,E m, and each end has an end neighborhood diffeomorphic to Σ Ei (0,1) for an orbifold Σ Ei. (Here, Σ Ei may not be uniquely determined up to diffeomorphisms except for some clear situations as here. ) By an exiting sequence of sets U i of O, we mean a sequence of neighborhoods {U i } so that U i p 1 O (K) /0 for only finitely many indices for each compact subset K of O. Each end neighborhood U diffeomorphic to Σ E (0,1) of an end E is covered by a connected open set Ũ in O where a subgroup of deck transformations ΓŨ acts on. Ũ is a component of p 1 O (U) = g π 1 (O) g(ũ). Each component of form g(ũ) is said to be a proper pseudo-end neighborhood. A pseudo-end sequence is an exiting sequence of proper pseudo-end neighborhoods U 1 U 2. Two pseudo-end sequences are compatible if an element of one sequence is contained eventually in the element of the other sequence. A compatibility class of a proper pseudo-end sequence is called a pseudo-end of O. Each of these corresponds to an end of O under the universal covering map p O. For a pseudo-end Ẽ of O, we denote by ΓẼ the subgroup ΓŨ where U and Ũ is as above. We call ΓẼ a pseudo-end fundamental group. A pseudo-end neighborhood U of a pseudo-end Ẽ is a ΓẼ-invariant open set containing a proper pseudo-end neighborhood of Ẽ. (See Proposition 1.4 also.) (From now on, we will replace pseudo-end with the abbreviation p-end.) p-end vertices. Let O be a real projective orbifold with the universal cover O. We fix a developing map dev in this subsection. Given a radial end of O and an end neighborhood U of a product form E [0,1) with a radial foliation, we take a component U 1 of p 1 (U) and the lift of the radial foliation. The developing images of leaves of the foliation end at a common point x in RP n. Recall that p-end vertex of O is the ideal point of leaves of U 1. (See ) When dev is fixed, we can identify it with its image under dev. It will be denoted by vẽ if its neighborhoods corresponds to a p-end Ẽ. Let S n 1 vẽ denote the space of equivalence classes of rays from vẽ diffeomorphic to an (n 1)-sphere where π 1 (Ẽ) acts as a group of projective automorphisms. Here, π 1 (Ẽ) acts on vẽ and sends leaves to leaves in U 1. Also, for a subset K of O, we denote by R vẽ (K) the space of directions of developed images of leaves in O under dev mapping to rays oriented away from vẽ ending at K. We have R vẽ (K) S n 1 vẽ.

53 38 2. PRELIMINARIES Given a p-end Ẽ corresponding to vẽ, we denote by R vẽ ( O) = ΣẼ the space of directions of developed leaves under dev oriented away from vẽ in O. The space develops to S n 1 x by dev as an embedding to a convex open domain. Recall that ΣẼ/ΓẼ is projectively diffeomorphic to the end orbifold to be denoted by Σ E or by ΣẼ. (See Lemma 1.6.) We may use the lifting of dev to S n. The end point x of the lift of radial lines will be identified with the p-end vertex also when the lift of dev is fixed. Here, we can canonically identify S n 1 x and S n 1 x and the group actions of ΓẼ on them p-end ideal boundary components. We recall Section Given a totally geodesic end of O and an end neighborhood U of the product from E [0,1) with a compactification by a totally geodesic orbifold E, we take a component U 1 of p 1 (U) and a convex domain SẼ, the ideal boundary component, developing to totally geodesic hypersurface under dev. Here Ẽ is the p-end corresponding to E and U 1. There exists a subgroup ΓẼ acting on SẼ. Again SẼ := SẼ/ΓẼ is projectively diffeomorphic to the end orbifold to be denote by S E or SẼ. We call SẼ a p-end ideal boundary component of O. We call SẼ an ideal boundary component of O. We may regard SẼ as a domain in a hyperspace in RP n or S n A Lie group invariant p-end neighborhoods. We need the following lemma later. LEMMA Let U be a p-r-end neighborhood of an end Ẽ where an end holonomy group ΓẼ acts on. Let Q be a discrete subgroup of ΓẼ. Suppose that G is a connected Lie group virtually containing Q so that G Q\G is compact. Assume that G acts on the p-r-end vertex vẽ and ΣẼ. Then g G g(u) contains a non-empty G-invariant p-end neighborhood of Ẽ. PROOF. We first assume that O S n and Q G. Let F be the compact fundamental domain of G under G Q. It is sufficiently to prove for the case when U is a proper p-end neighborhood since for any open set V containing U, g G g(v ) contains a p-end neighborhood g G g(u). Hence, we assume that bdu/γẽ is a smooth compact surface. Let F U denote the fundamental domain of bdu. Let F be a compact fundamental domain of G with respect to Q. Let L be a compact subset of ΣẼ and let ˆL U denote the union of all maximal open segments with endpoint vẽ in the direction of L. We claim that g F g(u) contains an open set in ˆL U. We show this by proving that g F g(u) l for any maximal l in ˆL U has a lower bounded on d-length. The lower bound is uniform for L. Suppose not. Then there exists sequence g i F and maximal segment l i in ˆL U so that g i (U) l i becomes smaller and smaller segment from vẽ as i. The endpoint of g i (U) l i equals g i (y i ) for y i bdu. This implies that g i (y i ) vẽ. Now, y i correspond to a direction u i ΣẼ. Since F is a compact set, u i corresponds to a point of a compact set F 1 (u). Hence, y i ˆF U for a compact subset ˆF U of bdu corresponding to the direction of F 1 (u). Since vẽ is a fixed point of G, and y i ˆF U for a compact subset ˆF U of S n not containing vẽ, this shows that g i form an unbounded sequence in SL ± (n + 1,R). This is a contradiction to g i F.

54 2.2. AFFINE ORBIFOLDS 39 We have a nonempty set Û := g(u) = g(u) g G g F containing an open set in U. G acts on Û clearly. We take the interior of Û. If G only virtually contains ΓẼ, we just need to add finitely many elements to the above arguments. [S n P] 2.2. Affine orbifolds An affine orbifold is an orbifold with a geometric structure modeled on (R n,aff(r n )). An affine orbifold has a notion of affine geodesics as given by local charts. Recall that a geodesic is complete in a direction if the affine geodesic parameter is infinite in the direction. An affine orbifold has a parallel end if the corresponding end has an end neighborhood foliated by properly embedded affine geodesics parallel to one another in charts and each leaf is complete in one direction. We assume that the affine geodesics are leaves assigned as above. We obtain a smooth complete vector field X E in a neighborhood of E for each end following the affine geodesics, which is affinely parallel in the flow; i.e., leaves have parallel tangent vectors. We call this an end vector field. We denote by X O the vector field partially defined on O by taking the union of vector fields defined on some mutually disjoint neighborhoods of the ends using the partition of unity. The oriented direction of the parallel end is uniquely determined in the developing image of each p-end neighborhood of the universal cover of O. Finally, we put a fixed complete Riemannian metric on O so that for each end there is an open neighborhood where the metric is invariant under the flow generated by X O. Note that such a Riemannian metric always exists. An affine orbifold has a totally geodesic end E if each end can be completed by a totally geodesic affine hypersurface. That is, there exists a neighborhood of the end E diffeomorphic to E [0,1) that compactifies to an orbifold diffeomorphic to E [0,1], and each point of E {1} has a neighborhood affinely diffeomorphic to a neighborhood of a point p in H for a half-space H of an affine space. This implies the fact that the corresponding p-end holonomy group h(π 1 (Ẽ)) for a p-end Ẽ going to E acts on a hyperspace P corresponding to E {1}. Recall that an orbifold is a topological space stratified by open manifolds (See Chapter 4 of [46]). An affine or projective orbifold is triangulated if there is a smoothly embedded n-cycle consisting of geodesic n-simplices on the compactified orbifold relative to ends by adding an ideal point to a radial end and an ideal boundary to each totally geodesic ends. where the interiors of i-simplices in the cycle are mutually disjoint and are embedded in strata of the same or higher dimension Affine suspension constructions. The affine subspace R n+1 is a dense open subset of RP n+1 which is the complement of (n + 1)-dimensional projective space RP n+1. Thus, an affine transformation is a restriction of a unique projective automorphism acting on R n+1. The group of affine transformations Aff(R n+1 ) is isomorphic to the group of projective automorphisms acting on R n+1 identified this way by the restriction homomorphism.

55 40 2. PRELIMINARIES A dilatation γ in an affine subspace R n+1 is a linear transformation with respect to an affine coordinate system so that all its eigenvalues have norm > 1 or < 1. Here, γ is an expanding map in the dynamical sense. A scalar dilatation is a dilation with a single eigenvalue. An affine orbifold O is radiant if h(π 1 (O)) fixes a point in R n+1 for the holonomy homomorphism h : π 1 (O) Aff(R n+1 ). A real projective orbifold O of dimension n has a developing map dev : O S n and the holonomy homomorphism h : π 1 (O) SL ± (n + 1,R). Here, S n is embedded as a unit sphere in R n+1. We obtain a radiant affine (n +1)-orbifold by taking O and dev and h : Define D : O R + R n+1 by sending (x,t) to tdev (x). For each element of γ π 1 (O), we define the transformation γ on O R + by (2.3) γ (x,t) =(γ(x),θ(γ) h (γ)(tdev (x)) ) for a homomorphism θ : π 1 (O) R +. Also, there is a transformation S s : O R + O R + sending (x,t) to (x,st) for s R +. Thus, O R + / S ρ,π 1 (O),ρ R +,ρ > 1 is an affine orbifold with the fundamental group isomorphic to π 1 (O) Z where the developing map is given by D the holonomy homomorphism is given by h and sending the generator of Z to S ρ. We call the result the affine suspension of O, which of course is radiant. The representation of π 1 (O) Z with the center Z mapped to a scalar dilatation is called an affine suspension of h. A special affine suspension is an affine suspension with θ 1 identically. There is a variation called generalized affine suspension. Here we use any γ that is a dilatation and normalizes h (π 1 (O)) and we deduce that O R + / γ,π 1 (O) is an affine orbifold with the fundamental group isomorphic to π 1 (O),Z. (See Sullivan- Thurston [130], Barbot [9] and Choi [41] also.) DEFINITION We denote by C( O) the manifold O R with the structure given by D, and say that C( O) is the affine suspension of O. Let S t : R n+1 R n+1, given by v t v, t R +, be a one-parameter family of dilations fixing a common point. A family of self-diffeomorphisms Ψ t on an affine orbifold M lifting to ˆΨ t : M M so that D ˆΨ t = S e t D for t R is called a group of radiant flow diffeomorphisms. LEMMA Let O be a strongly tame real projective n-orbifold. An affine suspension O of O always admits a group of radiant flow diffeomorphisms. Here, {Φ t } is a circle and all flow lines are closed. Conversely, if there exists a group of radiant flow diffeomorphisms where all orbits are closed and have the homology class [[ S P 1]] on O S 1 with an affine structure, then O S 1 is affinely diffeomorphic to one obtained by an affine suspension construction from a real projective structure on O. PROOF. The first item is clear by the above construction. The generator of π 1 (S 1 ) factor goes to a scalar dilatation. Thus, each closed curve along S 1 gives us a nontrivial homology. The homology direction of the flow equals

56 2.3. THE NEEDED LINEAR ALGEBRA 41 [[ S 1 ]] S(H 1 (O S 1 ;R)). By Theorem D of [67], there exists a connected crosssection homologous to [O ] H n (O S 1,V S 1 ;R) = H 1 (O S 1 ;R) where V is the union of the disjoint end neighborhoods of product form in O. By Theorem C of [67], any cross-section is isotopic to O. The radial flow is transversal to the crosssection isotopic to O and hence O admits a real projective structure. It follows easily now that O S 1 is an affine suspension. (See [9] for examples.) An affine suspension of a horospherical orbifold is called a suspended horoball orbifold. An end of an affine orbifold with an end neighborhood affinely diffeomorphic to this is said to be of suspended horoball type. This has also a parallel end since the parallel direction is given by the fixed point in the boundary of R n. PROPOSITION Under the affine suspension construction, a real projective n- orbifold has radial, totally geodesic, or horospherical ends if and only if the affine (n+1)- orbifold affinely suspended from it has parallel, totally geodesic, or suspended horospherical ends The needed linear algebra Here, we will collect the linear algebra we will need in this monograph. The main source is comprehensive Hoffman and Kunz [87]. DEFINITION Given an eigenvalue λ of an element g SL ± (n + 1,R), a C- eigenvector v is a nonzero vector in RE λ (g) := R n+1 ( ker(g λi) + ker(g λi) ),λ 0,Iλ 0 A C-fixed point is the direction of a C-eigenvector in RP n (resp. S n or CP n ). Any element of g has a primary decomposition. (See Section 6.8 of [87].) Write the minimal polynomial of g as m i=1 (x λ i) r i for r i 1 and mutually distinct complex numbers λ 1,...,λ m. Define C λi (g) := ker(g λ i I) r i C n+1 where r i = r j if λ i = λ j. Then the primary decomposition theorem states m C n+1 = C λi (g). i=1 A real primary subspace is the sum R n+1 (C λ (g) C λ (g)) for λ an eigenvalue of g. A point [ v], v R n+1, is affiliated with a norm µ of an eigenvalue if v R µ (g) := C λi (g) R n+1. i { j λ j =µ} Let µ 1,..., µ l denote the set of distinct norms of eigenvalues of g. We also have R n+1 = li=1 R µi (g). Here, R µ (g) {0} if µ equals λ i for at least one i. PROPOSITION Let g be an element of PGL(n + 1,R) (resp. SL ± (n + 1,R)) acting on RP n (resp. S n ). Let V and W be independent subspaces where g acts on. Suppose that the every norm of the eigenvalue of any eigenvector in the direction of V is strictly larger than any norms of the eigenvalues of the vectors in the direction of W. Let V S be the subspace that is the join of the C-eigenspaces of V. Then

57 42 2. PRELIMINARIES for x RP n Π(V ) Π(W) (resp. S n Π (V ) Π (W)), g n (x) accumulates to only points in Π(V S ) (resp. Π (V S )) as n. Let U be a neighborhood of x in RP n Π(V ) Π(W) (resp. S n Π (V ) Π (W)). Each point of an open subset of Π(V S ) (resp. Π (V S )) is realized a limit point of g n (y) as n for some y in U. PROOF. It is sufficient to prove for C n+1 and CP n. We write the minimal polynomial of g as m i=1 (x λ i) r i for r i 1 and mutually distinct complex numbers λ 1,...,λ m. Let W C be the complexification of the subspace corresponding to W and V C the one for V. Then C λi (g) is a subspace of W C or V C by elementary linear algebra. Now, we write the matrix of g determined only up to ±I in terms of above primary decomposition spaces. Then we write the matrix in the Jordan form in an upper triangular form. The nondiagonal terms of the matrix of g n is dominated by diagonal terms. The lemma easily follows. The last part follows by writing x and y in terms of vectors in directions of V and W and other g-invariant subspaces Nilpotent and orthopotent groups. Let U denote a maximal nilpotent subgroup of SL ± (n + 1,R) given by lower triangular matrices with diagonal entries equal to 1. LEMMA The matrix of g Aut(S n ) can be written under an orthogonal coordinate system as k(g)a(g)n(g) where k(g) is an element of O(n + 1), a(g) is a diagonal element, and n(g) is in the group U of unipotent lower triangular matrices. Also, diagonal elements of a(g) are the norms of eigenvalues of g as elements of Aut(S n ). PROOF. Let v 1,..., v n+1 denote the basis vectors of R n+1 that are chosen from the real Jordan-block subspaces of g with the same norms of eigenvalues. We require [ v 1 ] = vẽ. Now we fix a Euclidean metric on R n+1. We obtain vectors v 1,..., v n+1 by the Gram-Schmidt orthogonalization process using the corresponding Euclidean metric on R n+1. (See also Proposition 2.1 of Kostant [99].) We define U := k O(n+1) kuk 1. COROLLARY Let G be a subgroup of SL ± (n + 1,R). Suppose that we have for a positive constant C 1, and g G, 1 C 1 λ n+1 (g) λ 1 (g) C 1 for the minimal norm λ n+1 (g) of the eigenvalue of g and the maximal norm λ 1 (g) of the eigenvalues of g. Then g is in a bounded distance from U with the bound depending only on C 1. PROOF. Let us fix an Iwasawa decomposition SL ± (n + 1,R) = O(n + 1)D n+1 U for a unimodular diagonal group D n+1. By Lemma 2.21, we can find an element k O(n + 1) so that g = kk(g)k 1 ka(g)k 1 kn(g)k 1

58 2.3. THE NEEDED LINEAR ALGEBRA 43 where k(g) K,a(g) D n,n(g) U. Then kk(g)k 1 O(n+1) and ka(g)k 1 is uniformly bounded from I by a constant depending only on C 1 by assumption. Finally, we obtain kn(g)k 1 U. A subset of a Lie group is of polynomial growth if the volume of the ball B R (I) radius R is less than or equal to a polynomial of R. As usual, the metric is given by the standard positive definite left-invariant bilinear form that is invariant under the conjugations by the compact group O(n + 1). LEMMA U is of polynomial growth in terms of the distance from I. PROOF. Let Aut(S n ) have a left-invariant Riemannian metric. Clearly U is of polynomial growth by Gromov [81] since U is nilpotent. Given fixed g O(n + 1), the distance between gug 1 and u for u U is proportional to a constant c g, c g > 1, multiplied by d(u,i). Choose u U which is unipotent. We can write u(s) = exp(s u) where u is a nilpotent matrix of unit norm. g(t) := exp(t x) for x in the Lie algebra of O(n + 1) of unit norm. For a family of g(t) O(n + 1), we define (2.4) u(t,s) = g(t)u(s)g(t) 1 = exp(sad g(t) u). We compute 1 du(t,s) u(t,s) := u(t,s) 1 ( xu(t,s) u(t,s) x) = (Ad dt u(t,s) 1 I)( x). Since u is nilpotent, Ad u(t,s) 1 I is a polynomial of variables t,s. The norm of du(t,s)/dt is bounded above by a polynomial in s and t. The conjugation orbits of O(n+1) in Aut(S n ) are compact. Also, the conjugation by O(n + 1) preserves the distances of elements from I since the left-invariant metric µ is preserved by conjugation at I and geodesics from I go to geodesics from I of same µ-lengths under the conjugations by (2.4). Hence, we obtain a parametrization of U by U and O(n+1) where the volume of each orbit of O(n+1) grows polynomially. Since U is of polynomial growth, U is of polynomial growth in terms of the distance from I. THEOREM 2.24 (Zassenhaus [141]). For every discrete group G of GL(n, R), all of which have the shape in a complex basis in R n e iθ 1 0 e iθ e iθ n there exists a positive number ε, so that all the matrices A from G which satisfy the inequalities e iθ j 1 < ε for every j = 1,...,n are contained in the radical of the group, i.e., the subgroup G u of elements of G with only unit eigenvalues. An element g of GL(n+1, R) (resp. PGL(n+1, R)) is said to be unit-norm-eigenvalued if it (resp. its representative) has only eigenvalues of norm 1. A group is unit-normeigenvalued if all of its elements are unit-norm-eigenvalued. Recall that a subgroup of SL ± (n + 1,R) is orthopotent if there is a flag of subspaces 0 = Y 0 Y 1 Y m = R n+1 preserved by G so that G acts as an orthogonal group on Y j+1 /Y j for each j = 0,...,m 1. (See D. Fried [68].) THEOREM Let G be a unit-norm-eigenvalued subgroup of SL ± (n + 1,R). Then G is orthopotent, and the following holds:

59 44 2. PRELIMINARIES If G is discrete, then G is virtually unipotent. If G is a Lie group, then G is an extension of a solvable group by a compact group. If G is contractible, then G is a simply connected solvable Lie group. PROOF. By Corollary 2.22, G is in U. If G is discrete, then G is of polynomial growth. By Gromov [81], G is virtually nilpotent. Since G is solvable, Theorem of [133] shows that G can be put into an upper triangular form for a complex basis. The map G G/G u factors into a map G (S 1 ) n by taking the complex eigenvalues. By Theorem 2.24, the image is a discrete subgroup of (S 1 ) n. Hence, G/G u is finite where G u is unipotent. Suppose that G is a connected Lie group. Elements of G have only unit-norm eigenvalues. Since a(g) = I for a(g) in the proof of Lemma 2.21 for all g G, the proof of Corollary 2.22 shows that G O(n + 1)UO(n + 1) for a compact Lie group K. Since U is a distal group, G is a distal group, and hence G is orthopotent by [56] or [120]. By Corollary 2.1 of Jenkins [89], G is an extension of a solvable Lie group by a compact Lie group. If G is contractible, G then cannot be an extension by a compact group and must be solvable by the second item. Since connected solvable group is homotopy equivalent to a torus, we are done Elements of dividing groups. Suppose that Ω, Ω S n (resp. RP n ), is an open domain that is properly convex but not necessarily strictly convex. Let Γ, Γ SL ± (n + 1,R) (resp. PGL(n + 1,R), be a discrete group acting on Ω so that Ω/Γ is compact. An element of Γ is said to be elliptic if it is conjugate to an element of a compact subgroup of PGL(n + 1,R) or SL ± (n + 1,R). LEMMA Suppose that Ω is a properly convex domain, and Γ is a group of projective automorphisms of Ω. Suppose that Ω/Γ is an orbifold. Then an element g of Γ is elliptic if and only if g fixes a point of Ω if and only if g is of finite order. PROOF. Let us assume Ω S n. Let g be an elliptic element of Γ. Take a point x Ω. Let x denote a vector in a cone C(Ω) R n+1 corresponding to x. Let g denote the corresponding Then the orbits {g n ( x) n Z} has a compact closure. There is a fixed vector in C(Ω), which corresponds to a fixed point of Ω. If x is a point of Ω fixed by g, then it is in the stabilizer group. Since Ω/Γ is an orbifold, g is of finite order. If g is of finite order, g is certainly elliptic. [S n T] We recall the definitions of Benoist [18]: For an element g of SL ± (n+1,r), we denote by λ 1 (g),...,λ n (g) the sequence of the norms of eigenvalues of g with repetitions by their respective multiplicities. The first one λ 1 (g) is called the spectral radius of g. Assume λ 1 (g) λ n (g) for the following definitions. An element g of SL ± (n + 1,R) is proximal if λ 1 (g) has multiplicity one. g is positive proximal if g is proximal and λ 1 (g) is an eigenvalue of g. An element g of SL ± (n + 1,R) is semi-proximal if λ 1 (g) or λ 1 (g) is an eigenvalue of g. An element g of SL ± (n+1,r) is positive semi-proximal if λ 1 (g) is an eigenvalue of g. g is called positive bi-semi-proximal if g and g 1 is both positive semi-proximal.

60 2.3. THE NEEDED LINEAR ALGEBRA 45 g is called positive bi-proximal if g and g 1 is both positive proximal. Of course, the proximality is a stronger condition than semi-proximality. For each bi-semi-proximal element g Γ, we have two disjoint compact convex subspaces A g := A Cl(Ω) and R g := R Cl(Ω) for the eigenspace A associated with the largest of eigenvalues of g and the eigenspace R associated with the smallest of the eigenvalues of g. Note g A g and g R g are both identity maps. Here, A g is associated with λ 1 (g) and R g is with λ n (g), which is an eigenvalue as well. A g is called an attracting fixed subset and R g a repelling fixed subset. Let g be a bi-semi-proximal element. For g, g SL ± (n + 1,R), we denote by V A g := ker(g λ 1 (g)i) m 1 : R n+1 R n+1 where m 1 is the multiplicity of the eigenvalue λ 1 (g) in the characteristic polynomial of g. We denote by V R g = ker(g λ n (g)i) m n : R n R n+1 where m n is the multiplicity of the eigenvalue λ n (g). We denote  g = V A g Cl(Ω) and ˆR g = V R g Cl(Ω). Clearly, A g  g and R g ˆR g. LEMMA Suppose g acts on a properly convex domain with an eigenvalue of norm > 1. Then g is positive bi-semi-proximal. PROOF. Since g acts on a proper cone in R n+1 corresponding to Ω, the eigenvalue corresponding to A g is positive by Lemma 5.3 of [18], and so are one corresponding to R g. This proves that g is bi-semi-proximal. We generalize Proposition 5.1 of Benoist [17]. By Theorem 2.11 following from Selberg s Lemma [126], there is a finite index subgroup Γ Γ elements of Γ are not elliptic. (In fact a finite manifold cover is enough.) THEOREM 2.28 (Benoist [18]). Suppose that Ω is properly convex but not necessarily strictly convex. Let Γ be a discrete group acting on Ω so that Ω/Γ is compact and Hausdorff. Let Γ be the finite index subgroup of Γ without torsion. Then each nonidentity element g, g Γ is bi-semi-proximal. A g,â g bdω, R g, ˆR g bdω are properly convex subsets in the boundary. dima g = dimker(g λ 1 I) 1 and dimr g = dimker(g λ n I) 1. Let K be a compact set in Ω. Then {g i (K) n 0} has the limit set in A g, and {g i (K) n < 0} has the limit set in R g. Furthermore, if Ω is strictly convex, then A g =  g is a point in bdω and R g = ˆR g is a point in bdω and g is positive bi-proximal. PROOF. By Proposition 2.29, every nonidentity element g of Γ has a norm of eigenvalue > 1. By Lemma 2.27, g is positive bi-semi-proximal. By Proposition 2.20, A g is a limit point of {g i (x) i > 0}. Hence, A g is not empty and A g bdω. Similarly R g is not empty as well. Then A g equals the intersection V 1 Cl(Ω) for the eigenspace V 1 of g corresponding to λ 1 (g). Since g fixes each point of V 1, it follows that A g is a compact convex subset of bdω. Similarly, R g is a compact convex subset of bdω. Suppose that  g Ω /0. Then g acts on the open properly convex domain  g Ω as a unit-norm-eigenvalued element. By following the proof of Proposition 2.29, we obtain a contradiction again by obtaining curves of d Ω -lengths that can be made as arbitrarily short. Thus,  g bdω. As above, it is a compact convex subset. Similarly, ˆR g is a compact convex subset of bdω.

61 46 2. PRELIMINARIES The second item follows from the second item of Proposition Suppose that Ω is strictly convex. Then dima g = 0,dim g = 0,dimR g = 0,dim ˆR g = 0 by the strict convexity. Proposition 5.1 of [17] proves that g is proximal. g 1 is also proximal by the same proposition These are positive proximal since g acts on a proper cone. Hence, g is positive bi-proximal. Note here that  g may contain A g properly and ˆR g may contain R g properly also. PROPOSITION Suppose that Γ is a discrete group acting on Ω so that Ω/Γ is compact and Hausdorff. Let g be a non-torsion element as in Theorem Then g has some norm of eigenvalues > 1. Furthermore, g does not act with a single norm of eigenvalues on any subspace Q with Q Ω /0. In particular g is not orthopotent nor unipotent. PROOF. Suppose that g acts with a single norm of eigenvalues on a subspace Q with Q Ω /0. It will be sufficient to show that this is a contradiction since Q can be the whole S n and by Theorem Applying Lemma 2.30 where n is replaced by the dimension of Q, we obtain 0 as the infimum of the Hilbert lengths of closed curves in a compact orbifold Ω/Γ. Since Ω/Γ is a compact orbifold, there should be a positive lower bound. This is a contradiciton. LEMMA Suppose that Ω is a properly convex domain in A n. Suppose that an element g acts on Ω with only single norm of eigenvalue. Then inf {d Ω(y,g(y)) y Ω Q} = 0. y Ω PROOF. g fixes a point x in Cl(Ω) by the Brouwer-fixed-point theorem. By Lemma 2.26, g does not fix a point in Ω. Assume x bdω Q is a fixed point of g. We will obtain inf {d Ω(y,g(y)) y Ω Q} = 0. y Ω This will show that Ω/Γ contains a non-null-homotopic closed loop of arbitrarily short length corresponding to g. Since Ω/Γ is a closed orbifold, this is absurd for a compact space with a Finsler path-metric. (See Kobayashi [97].) We may assume that Ω A n for an affine space A n since Ω is properly convex. We choose a coordinate system where x is the origin of A n. Then g has a form of a rational map. We denote by Dg x the linear map that is the differential of g at x. Let S r denote the similarity transformation of A n fixing x. Then we obtain S r g S 1/r : S r (Ω) S r (g(ω)). Recall the definition of Dg x : R n R n is one satisfying g(y) g(u) Dg x (y u) lim 0. y,u 0 y u Hence, lim r g(s1/r (y)) g(s 1/r (u)) S 1/r Dg x (y u) 0. r Setting u = 0, we obtain lim S r g S 1/r (y) Dg x (y) 0. r

62 2.3. THE NEEDED LINEAR ALGEBRA 47 We obtain that as r, S r gs 1/r converges to Dg x on a sufficiently small open ball around x. Also, it is easy to show that as r, S r (Ω) geometrically converges to a cone Ω x, with the vertex at x on which Dg x acts on. Let x n be an affine coordinate function for a sharply-supporting hyperspace of Ω taking 0 value at x. It will be specified a bit later. For now any such one will do. Let x(t) be a projective geodesic with x(0) = x at t = 0 and x(t) Ω,x n (x(t)) = t for t > 0 and let u = dx(t)/dt 0 at t = 0. We assume that g is unit-norm-eigenvalued. Then lim d Ω(g(x(t)),x(t)) = d Ωx, (Dg x ( u), u) t 0 considering u as an element of the cone Ω x, : This follows from d Ω (g S 1/r S r (x(t)),x(t)) = d Sr (Ω)(S r g S 1/r (S r (x(t))),s r (x(t))) since S r : (Ω,d Ω ) (S r (Ω),d Sr (Ω)) is an isometry. We set x(t) = x(1/r) and obtain S r (x(1/r)) u as r. Since S r (Ω) Ω x, as r, (2.2) shows that (2.5) d Sr (Ω)(S r g S 1/r (S r (x(1/r))),s r (x(1/r))) d Ωx, (Dg x ( u), u). Now, Ω o x, is a convex cone of form C(U) for a convex open domain U in the infinity of A n. The set U of sharply-supporting hyperspaces of Ω o x, at x is a convex compact set whose interior is dual to U. We identify A n with a vector space by setting x to be the origin. Since g acts on a ball U, g fixes a point by the Brouwer-fixed-point theorem, which corresponds to a hyperspace. Let P be a hyperspace in A n passing x sharply-supporting Ω o x, invariant under Dg x. Let x n denote the affine coordinate function where P is given by x n = 0 and points of Ω o x, satisfy x n > 0. First, suppose that Ω x, Q is properly convex. Projecting Ω o x, Q to the lowerdimensional space ˆQ S(A n {x}), we obtain the properly convex domain Ω 1. By the induction hypothesis on dimension dimω 1, since the Dg x -action on ˆQ has only one-norm of the eigenvalues, we can find a sequence z i Ω 1 so that d Ω1 (Dg x (z i ),z i ) 0. Since Ω x, is a proper convex cone in A n, we choose a sequence u i Ω x, Q with x n (u i ) = 1 and u i has the direction of z i from x. Let u i denote the vector xz i on Q A n where x n ( u i ). Since g is unit-norm-eigenvalued in Q, x n (Dg x ( u i )) = 1 also. Hence, the geodesic to measure the Hilbert metric from u i to Dg x ( u i ) is on x n = 1. Let P 1 denote the plane x n = 1. The projection from x from Ω x, P 1 U is a projective diffeomorphism and hence is an isometry. Therefore, d Ωx, (Dg x ( u i ), u i ) 0. We can find arcs x i (t) with x n (x i (t)) = t and dx i (t)/dt = u i at t = 0. Also, we find a sequence of points x i (t i ), t i 0, so that d Ω (g(x i (t i )),x i (t i )) = d S1/ti (Ω)(S 1/ti g S ti (S 1/ti (x i (t i ))),S 1/ti (x i (t i ))). Since S 1/t j (Cl(Ω)) Cl(Ω x, ), and S 1/t j (x i (t j )) u i as j, we obtain d Ω (g(x i (t j )),x i (t j )) d Ωx, (Dg x ( u i ), u i ) by (2.5). By choosing j i sufficiently large for each i, we obtain d Ω (g(x i (t ji )),x i (t ji )) 0,

63 48 2. PRELIMINARIES Suppose that Ω x, is not a properly convex cone. The projection of this set to S(A n {x}) is a convex but not properly convex set. By Proposition 2.8, such a set is foliated by complete affine spaces of dimension j, j > 0. Hence, Ω x, is foliated by complete affine half-spaces of dimension j + 1. The quotient space O x := Ω x, / with equivalence relationship given by the foliation is a properly convex open domain of dimension < n. Recall that there is a pseudo-metric d Ωx, on Ω x,. Note that the projection π : Ω x, O x is projective and d Ωx, (y,z) = d Ox (π(y),π(z) for all x,y d Ωx,, which is fairly easy to show. The differential Dg x induces a linear map Dg x : O x O x. By the induction, for any ε > 0, g acts on O x so that d Ox (y,dg x(y)) < ε for a point y O x. We take an inverse image y in Ω x, of y. Then d Ωx, (y,dg x (y)) = d Ox (y,dg x(y)) < ε. Since g x is unipotent, we have x n (Dg x ( u )) = 1,x n ( u ) = 1 as above. Since A is an affine half-space, we obtain d Ωx, (Dg x ( u ), u ) = 0. (See Koszul [100].) Similarly to above, we obtain d Ω (g(x i (t ji )),x i (t ji )) 0 for some t ji 0 and x i chosen as above The higher-convergence-group. For this section, we only work with S n since only this version is needed. We considering SL ± (n+1,r) as an open subspace of M n+1 (R). We can compactify SL ± (n+1,r) as S(M n+1 (R)). Denote by ((g)) the equivalence class of g SL ± (n + 1,R). THEOREM 2.31 (The higher-convergence-group property). Let g i be any unbounded sequence of projective automorphisms of a properly convex domain Ω. Then we can choose a subsequence of ((g i )) converging to ((g )) in S(M n+1 (R)) for g M n+1 (R) where the following hold : g is undefined on S(kerg ) and the range is S(Img ). dims(kerg ) + dims(img ) = n 1. For every compact subset K of S n S(kerg ), g i (K) K for a subset K of S(Img ). Given a convergent subsequence g i as above, ((gg i )) is also convergent to ((gg )) and S(kergg ) = S(kerg ) and S(Imgg ) = gs(g ) ((g i g)) is also convergent to ((g g)) and S(kerg g) = g 1 (S(kerg )) and S(Img g) = S(g ). PROOF. Since S(M n+1 (R)) is compact, we can find a subsequence of g i converging to an element ((g )). The second item is the consequence of the rank and nullity of g. The third item follows by considering the compact open topology of maps and g i divided by its maximal norm of the matrix entries. The two final item are easy to see. LEMMA g can be obtained by taking the limit of g i /m(g i ) in M n+1 (R) where m(g i ) is the maximal norm of elements of g i in the matrix form of g i. Also, we can use g i /λ 1 (g i ) for the maximal norm λ 1 (g i ) of the eigenvalue of g i. PROOF. This follows since g i /m(g i ) does not go to zero. Also, it is easy to see (2.6) m(g) λ 1 (g) nm(g).

64 2.3. THE NEEDED LINEAR ALGEBRA 49 DEFINITION A sequence {g i }, g i SL ± (n + 1,R) so that ((g i )) is convergent in S(M n+1 (R)) is called a convergence sequence. In the above g M n+1 (R) is called a convergence limit, determined only up to a positive constant. We may also do this for PGL(n+1,R). A sequence {g i }, g i PGL(n+1,R) so that g i is convergent in P(M n+1 (R)) is called a convergence sequence. The element g M n+1 (R) where {((g i ))} ((g )) is called called a convergence limit, determined only up to a nonzero constant. Given a convergence sequence g i, we define (2.7) (2.8) (2.9) (2.10)  ({g i }) := S(Img ) ˆN ({g i }) := S(kerg ) A ({g i }) := S(Img ) Cl(Ω) N ({g i }) := S(kerg ) Cl(Ω) We use the KAK-decomposition (or polar decomposition) of Cartan for SL ± (n+1,r). We may write g i = k i d iˆk i 1 where k i, ˆk i O(n + 1,R) and d i is a positive diagonal matrix with a nonincreasing set of elements a 1,i a 2, j a n+1, j. Let S([1,m]) the subspace spanned by e 1,..., e m and S([m+1,n+1]) the subspace spanned by e m+1,..., e n+1. We assume that {k i } converges to k and {[d 1,...,d n+1 ]} is convergent in RP n. Recall Definition 2.19, we obtain THEOREM Let g i be a convergence sequence. Then there exists m a, 1 m a < n + 1, where a j,i /a 1,i 0 for j > m a and a j,i /a 1,i > ε for j m a for a uniform ε > 0. there exists m r, 1 m a < m r n + 1, where a j,i /a n+1,i < C for j m r for a uniform C > 1, and a j,i /a n+1,i for j < m r. ˆN ({g i }) is the geometric limit of ˆk i (S([m a + 1,n + 1])).  ({g i }) is the geometric limit of k i (S([1,m a ])). g 1 i is also a convergent sequence up to a choice of subsequences, and  ({g 1 i }) ˆN ({g i }) and ˆN ({g 1 i })  ({g i }). PROOF. These are straight forward. The last item follows by considering the second and third items. We define for each i, A p (g i ) := k i S([1,m a ]),N p (g i ) := ˆk i (S[m a + 1,n + 1]), F p (g i ) := k i S([1,m r 1]), and R p (g i ) := ˆk i S([m r,n + 1]). We define ˆR ({g i }) as the geometric limit of {R p (g i )}, and ˆF ({g i }) as the geometric limit of {F p (g i )}. We also define R ({g i }) := ˆR ({g i }) Cl(Ω),F ({g i }) := ˆF ({g i }) Cl(Ω).

65 50 2. PRELIMINARIES PROPOSITION A ({g i }) contains an open subset of  ({g i }) and hence A ({g i }) =  ({g i }). Also, R ({g i }) contains an open subset of ˆR ({g i }) and hence R ({g i }) = ˆR ({g i }). PROOF. We write g i = k i D iˆk i 1. By Theorem 2.34,  ({g i }) is the geometric limit of k i (S[1,m a ]) for some m a as above. g i (U) = k i D i (V i ) for an open set U O and V i = ˆk i 1 (U). Since ˆk i 1 is a d-isometry, V i is an open set containing a closed ball B i of fixed radius ε. D i converges to a diagonal matrix D. We may assume without loss of generality that B i B where B is a ball of radius ε. We may assume B i B B for a fixed ball of radius ε/2 for sufficiently large i. Then D i (B) D (B) S([1,m a ]). Here, D (B) is a subset of S([1,m a ]) containing an open set. Since D i (B i ) geometrically converges to a subset containing D (B), up to a choice of subsequence. Thus, k i D i (V i ) geometrically converges to a subset containing k D (B) by Lemma 2.3. LEMMA Suppose that Γ acts properly discontinuously on a properly convex open domain Ω and g i is a sequence in Γ. Suppose that g i is not bounded in SL ± (n+1,r). Then the following hold: (i) ˆR ({g i }) Ω = /0, (ii)  ({g i }) Ω = /0, (iii) ˆF ({g i }) Ω = /0, and (iv) ˆN ({g i }) Ω = /0. PROOF. (ii) Suppose not. Since  ({g i }) Ω /0, A ({g i }) meets Ω. Since A ({g i }) is the set of points of limits g i (x) for x Ω, the proper discontinuity of the action of Γ shows that A ({g i }) does not meet Ω. (iv) For each x in Ω, a fixed ball B in Ω centered at x does not meet ˆk i (S([m a + 1,n + 1])) for infinitely many i. Otherwise g m i (B) converges to a nonproperly convex set in Cl(Ω) as m, a contradiction. Hence, the second item follows. The remainder follows similarly up to changing g i to g 1 i. THEOREM Let g i be a convergence sequence in Γ acting properly discontinuously on a properly convex domain Ω. Then (2.11) (2.12) (2.13) (2.14) A ({g i }) =  ({g i }) Cl(Ω) =  ({g i }) bdω, N ({g i }) = ˆN ({g i }) Cl(Ω) = ˆN ({g i }) bdω, R ({g i }) = ˆR ({g i }) Cl(Ω) = ˆR ({g i }) bdω, F ({g i }) = ˆF ({g i }) Cl(Ω) = ˆF ({g i }) bdω are subsets of bdω and they are nonempty sets. Also, we have (2.15) (2.16) (2.17) (2.18)  ({g i }) ˆF ({g i }), A ({g i }) F ({g i }), ˆR ({g i }) ˆN ({g i }), R ({g i }) N ({g i }). PROOF. By Lemma 2.36, we only need to show the respective sets are not empty. By the third item of Theorem 2.31, a point x in  ({g i }) Cl(Ω) is a limit of g i (y) for some y Ω. Since Γ acts properly discontinuously, x Ω and x bdω. By taking {g 1 i }, we obtain ˆR ({g i }) Cl(Ω) /0. Since ˆR ({g i }) ˆN ({g i }) and  ({g i }) ˆF ({g i }), the rest follows.

66 The last collections are from definitions CONVEXITY AND CONVEX DOMAINS 51 PROPOSITION For an automorphism g of Ω, we have up to a choice of convergent subsequences (2.19) (2.20) (2.21) (2.22) (2.23)  ({gg i }) = g(â ({g i })), ({g i g}) =  ({g i }), ˆN ({gg i }) = ˆN ({g i }), ˆN ({g i g}) = g 1 ( ˆN ({g i })), ˆF ({gg i }) = g( ˆF ({g i })), ˆF ({g i g}) = ˆF ({g i }), ˆR ({gg i }) = ˆR ({g i }), ˆR ({g i g}) = g 1 ( ˆR ({g i })), ˆF ({g i }) = ˆN ({g 1 i }), ˆR ({g i }) =  ({g 1 i }) PROOF. The fourth and fifth items of Theorem 2.31 imply the first and second lines here. We have g 1 i We obtain = ˆk i di 1 k i where di 1 has diagonal entries a 1 i,n+1 a 1 i,n a 1 i,1. F p (g i ) = k i (S[1,m r 1]) = N p (g 1 i ),R p (g i ) = ˆk i (S[m r,n + 1]) = A p (g 1 i ). Hence, the final line holds. The third line follows from the second line by the fifth line. Also, the fourth line follows from the first line by the fifth line. Of course, there are RP n -versions of the results here. However, we do not state these Convexity and convex domains Convexity. In the following, if i = 1, then S 0 is a pair of antipodal points. PROPOSITION A real projective n-orbifold is convex if and only if the developing map sends the universal cover to a convex domain in RP n (resp. S n ). A real projective n-orbifold is properly convex if and only if the developing map sends the universal cover to a precompact properly convex open domain in an affine patch of RP n (resp. S n ). If a convex real projective n-orbifold is not properly convex and not complete affine, then its holonomy is reducible in PGL(n + 1,R) (resp. SL ± (n + 1,R)). In this case, O is foliated by affine subspaces l of dimension i with the common boundary Cl(l) l equal to a fixed subspace RP i 1 (resp. S i 1 ) in bdo. PROOF. We prove for S n first. Since a convex domain is projectively diffeomorphic to a convex domain in an affine space as defined in Section 1.1.1, the developing map must be an embedding since any local chart extends to a global embedding of an affine space into S n. The converse is also trivial. (See Proposition A.1 of [40]. ) The second follows immediately. For the final item, a convex subset of S n is a convex subset of an affine subspace A n, isomorphic to an affine space, which is the interior of a hemisphere H. We may assume that D o /0 by restricting to a spanning subspace of D in S n. Let D be a convex subset of H o. If D is not properly convex, the closure Cl(D ) must have a pair of antipodal points in H. They must be in bdh. A great open segment l must connect this antipodal pair in

67 52 2. PRELIMINARIES bdh and pass an interior point of D. If a subsegment of l is in bdd, then l is in a sharply supporting hyperspace and l does not pass an interior point of D. Thus, l D. Hence, D contains a complete affine line. Thus, D contain a maximal complete affine subspace. Two such complete maximal affine subspaces do not intersect since otherwise a larger complete affine subspace of higher dimension is in D by convexity. We showed in [29] that the maximal complete affine subspaces foliated the domain. (See also [63].) The foliation is preserved under the group action since the leaves are lower-dimensional complete affine subspaces in D. This implies that the boundary of the affine subspaces is a lower dimensional subspace. These subspaces are preserved under the group action. Hence, the holonomy group is reducible. For the RP n -version, we use the double covering map S n RP n mapping an open hemisphere to an affine subspace. [S n T] PROPOSITION Let Ω be a properly convex domain in S n. The image Ω be the image of Ω under the double covering map S n RP n. Then the restriction Cl(Ω) Cl(Ω ) is one-to-one and onto. PROOF. This follows since we can find an affine subspace A n containing Cl(Ω). Since the covering map restricts to a homeomorphism on A n, this follows. DEFINITION Given a convex set D in RP n, we obtain a connected cone C(D) in R n+1 {O} mapping to D, determined up to the antipodal map. For a convex domain D S n, we have a unique domain C(D) R n+1 {O}. A join of two properly convex subsets A and B in a convex domain D of RP n (resp. S n ) is defined as A B := {[t x + (1 t) y] x, y C(D),[ x] A,[ y] B,t [0,1]} (resp. A B := {((t x + (1 t) y)) x, y C(D),(( x)) A,(( y)) B,t [0,1]}) where C(D) is a cone corresponding to D in R n+1. The definition is independent of the choice of C(D). In RP n, the join may depend on D. Note we use p B = {p} B interchangeably for a point p. The join above does depend on the choice of cones. DEFINITION Let C 1,...,C m respectively be cones in a set of independent vector subspaces V 1,...,V m of R n+1. In general, a sum of convex sets C 1,...,C m in R n+1 in independent subspaces V i is defined by C 1 + +C m := {v v = c c m, c i C i }. A strict join of convex sets Ω i in S n (resp. in RP n ) is given as Ω 1 Ω m := Π(C 1 + +C m ) (resp. Π (C 1 + +C m )) where each C i {O} is a convex cone with image Ω i for each i for the projection Π (resp. Π ) The flexibility of boundary. The following lemma gives us some flexibility of boundary. A smooth hypersurface embedded in a real projective manifold is called strictly convex if under a chart to an affine subspace, it maps to a hypersurface which is defined by a real function with positive Hessians at points of the hypersurface. LEMMA Let M be a strongly tame properly convex real projective orbifold with strictly convex M. We can modify M inward M and the result bound a strongly tame or compact properly convex real projective orbifold M with strictly convex M

68 2.4. CONVEXITY AND CONVEX DOMAINS 53 PROOF. Let Ω be a properly convex domain covering M. We may assume that Ω S n. We may modify M by pushing M inward. We take an arbitrary inward vector field defined on a tubular neighborhood of M. (See Section 4.4 of [46] for the definition of the tubular neighborhoods.) We use the flow defined by them to modify M. By the C 2 -convexity condition, for sufficiently small change the image of M is still strictly convex and smooth. Let the resulting compact n-orbifold be denoted by M. M is covered by a subdomain Ω in Ω. Since M is a compact suborbifold of M, Ω is a properly embedded domain in Ω and thus, bdω Ω = Ω. Ω is a strictly convex hypersurface since so is M. This means that Ω is locally convex. A locally convex closed subset of a convex domain is convex by Lemma Hence, Ω is convex and hence is properly convex being a subset of a properly convex domain. So is M. [SS n ] REMARK Thus, by choosing one in the interior, we may assume without loss of generality that a strictly convex boundary component can be pushed out to a strictly convex boundary component Needed lemmas. The following will be used in many times in the monograph. LEMMA 2.45 (Chapter 11 of [131]). Let K be a closed subset of a convex domain Ω in RP n (resp. S n ) so that each point of bdk has a convex neighborhood. Then K is a convex domain. PROOF. We can connect each pair of points by a broken projective geodesics. Then local convexity shows that we can make the number of geodesic segments to go down by one using triangles. Finally, we may obtain a geodesic segment connecting the pair of points. [SS n ] LEMMA Let O be a strongly tame properly convex real projective orbifold. Let σ be a convex domain in Cl( O) P for a subspace P. Then either σ bdo or σ o is in O. PROOF. Suppose that σ o meets bdo and is not contained in it entirely. Since the complement of σ o bdo is a relatively open set in σ o, we can find a segment s σ o with a point z so that a component s 1 of s {z} is in bdo and the other component s 2 is disjoint from it. We may perturb s in a 2-dimensional totally geodesic space containing s and so that the new segment s Cl( O) meets bdo only in its interior point. This contradicts the fact that O is convex by Theorem A.2 of [40]. [SS n ] The Benoist theory. In late 1990s, Benoist more or less completed the theory of the divisible action as started by Benzécri, Vinberg, Koszul, Vey, and so on in the series of papers [17], [16], [18], [19], [13], [12]. The comprehensive theory will aid us much in this paper. PROPOSITION 2.47 (Corollary 2.13 [18]). Suppose that a discrete subgroup Γ of SL ± (n,r) (resp. PGL(n,R)), n 2, acts on a properly convex (n 1)-dimensional open domain Ω in S n 1 (resp, RP n 1 ) so that Ω/Γ is compact. Then the following statements are equivalent. Every subgroup of finite index of Γ has a finite center. Every subgroup of finite index of Γ has a trivial center. Every subgroup of finite index of Γ is irreducible in SL ± (n,r) (resp. in PGL(n,R)). That is, Γ is strongly irreducible.

69 54 2. PRELIMINARIES The Zariski closure of Γ is semisimple. Γ does not contain an infinite nilpotent normal subgroup. Γ does not contain an infinite abelian normal subgroup. PROOF. Corollary 2.13 of [18] considers PGL(n,R) and RP n 1. However, the version for S n 1 follows from this since we can always lift a properly convex domain in RP n 1 to one Ω in S n 1 and the group to one in SL ± (n,r) acting on Ω. The center of a group G is denoted by Z(G). A virtual center of a group G is a subgroup of G centralizing a finite index subgroup of G. The group with properties above is said to be the group with a trivial virtual center. THEOREM 2.48 (Theorem 1.1 of [18]). Let n 1 1. Suppose that a virtual-centerfree discrete subgroup Γ of SL ± (n,r) (resp. PGL(n,R)) acts on a properly convex (n 1)-dimensional open domain Ω S n 1 so that Ω/Γ is compact and Hausdorff. Then every representation of a component of Hom(Γ,SL ± (n,r)) (resp. Hom(Γ,PGL(n,R))) containing the inclusion representation also acts on a properly convex (n 1)-dimensional open domain cocompactly. When Ω/Γ admits a hyperbolic structure and n = 3, Inkang Kim [96] proved this simultaneously. PROPOSITION 2.49 (Theorem 1.1. of Benoist [16]). Assume n 2. Let Σ be a closed (n 1)-dimensional strongly tame properly convex projective orbifold, and let Ω denote its universal cover in S n 1 (resp. RP n 1 ). Then Ω is projectively diffeomorphic to the interior of a strict join K := K 1 K l0 where K i is a properly convex open domain of dimension n i 0 in the subspace S n i in S n (resp. RP n i in RP n ). K i corresponds to a convex cone C i R ni+1 for each i. Ω is the image of C 1 + +C l0. Let Γ i be the image of the restriction map of the subgroup of π 1(Σ) acting on K i. We denote by Γ i an arbitrary extension of Γ i by requiring it to act trivially on K j for j i. R l0 1 denote the diagonalizable group acting trivially on each K i. The fundamental group π 1 (Σ) is virtually isomorphic to a subgroup of R l0 1 Γ 1 Γ l0 for (l 0 1) + l 0 i=1 n i = n. π 1 (Σ) acts on K o cocompactly and discretely and semisimply (Theorem 3 of Vey [134]). Each Γ j acts on K j cocompactly, the Zariski closure G j is an irreducible reductive group, and G j acts trivially on K m for m j. The subgroup corresponding to R l0 1 acts trivially on each K j and form a diagonal matrix group. A virtual center of π 1 (Σ) of maximal rank is isomorphic to Z l0 1 corresponding to the subgroup of R l0 1. These have positive diagonalizable matrices and restricts to the identity on each K i. (Proposition 4.4 of [16].) The group indicated by Z l 0 1 is a virtual center of π 1 (Σ). See Example of Morris [121] for a group acting properly on a product of two hyperbolic spaces but restricts to a non-discrete group for each factor space. COROLLARY Assume as in Proposition Then a ( resp. virtually) normal solvable subgroup of Γ is ( resp. virtually) central.

70 2.4. CONVEXITY AND CONVEX DOMAINS 55 PROOF. If Γ is virtually abelian, this is obvious. Suppose that Ω is properly convex. Let G be a normal solvable subgroup of Γ. G is a normal solvable subgroup of the Zariski closure Z (Γ). By Theorem 1.1 of [16], Z (Γ) equals G 1 G l R l 1 + and K = K 1 K l where G i is reductive and the following holds: if K i is homogeneous, then G i is semi-simple and G i is commensurable with Aut(K i ). Otherwise, K o i is divisible and G i is a union of components of SL ± (V i ) The image of G into G i by the restriction homomorphism to K i is a virtually normal solvable subgroup of G i. Since G i is reductive, the image is the trivial group. Hence, G must be a subgroup of R l 1 + and hence is central. If l 0 = 1, Γ is strongly irreducible as shown by Benoist. However, the images of these groups will be subgroups of PGL(m,R) and SL ± (m,r) for m n. If l 0 > 1, we say that such an image in Γ is virtually factorable. Otherwise, such an image is non-virtually-factorable group. PROPOSITION Assume as in Proposition Then K is the closure of the convex hull of g Z l 0 1 A g for the attracting limit set A g of g not in the torsion subgroup. Also, for any partial join ˆK := K i1 K i j for a subcollection {i 1,...,i j }, the closure of the convex hull of g Z l 0 1 A g ˆK equals ˆK. PROOF. We take a finite-index normal subgroup Γ of Γ so that Z l is the center of Γ. Using Theorem 2.11, we may assume that Γ is torsion-free. Note that ka g for any k Γ equals A kgk 1 = A g since kgk 1 = g. Thus, Γ acts on g Z l 0 1 A g since it is a Γ - invariant set. The interior C of the convex hull of g Z l 0 1 A g is a subdomain in K o. Since C/Γ K o /Γ is a homotopy equivalence of closed manifolds, we obtain C = K o and Cl(C) = K. For the second part, if the closure of the convex hull of g Z l 0 1 A g ˆK is a proper subset of ˆK, then the closure of the convex hull of g Z l 0 1 A g is a proper subset of K. This is a contradiction. [SS n ] We have a following useful result. COROLLARY Let h i : Γ SL ± (n,r) be a sequence of faithful discrete representations so that O i := Ω i /h i (Γ) is a closed real projective orbifold for a properly convex domain Ω i. Suppose that h i h algebraically, and Cl(Ω i ) geometrically converges to a properly compact domain Cl(Ω ) with nonempty interior Ω. Then h (Γ) acts on the interior Ω so that the following hold: Ω /h (Γ) is a closed real projective orbifold. Ω is a convex domain. Ω /h (Γ) is diffeomorphic to O i for sufficiently large i. If U is a properly convex domain where h (Γ) acts so that U/h (Γ) is an orbifold, then U = Ω or J(Ω ) where J is a projective automorphism commuting with h (Γ). In particular, if Γ is non-virtually-factorable, then J = I or A. PROOF. See the proof of Theorem 4.1 of [53].

71 56 2. PRELIMINARIES Technical propositions. By the following, the first assumption of Theorem 6.31 are needed only for the conclusion of the theorem to hold. PROPOSITION If a group G of projective automorphisms acts on a strict join A = A 1 A 2 for two compact convex sets A 1 and A 2 in S n (resp. RP n ), then G is virtually reducible. PROOF. We prove for S n. Let x 1,...,x n+1 denote the homogeneous coordinates. There is at least one set of strict join sets A 1,A 2. We choose a maximal number collection of compact convex sets A 1,...,A m so that A is a strict join A 1 A m. Here, we have A i S i for a subspace S i corresponding to a subspace V i R n+1 that form an independent set of subspaces. We claim that g G permutes the collection {A 1,...,A m}: Suppose not. We give coordinates so that for each i, there exists some index set I i so that elements of A i satisfy x j = 0 for j I i and elements of A satisfy x i 0. Then we form a new collection of nonempty sets J := {A i g(a j) 0 i, j n,g G} with more elements. Since A = g(a) = g(a 1) g(a l ), using coordinates we can show that each A i is a strict join of nonempty sets in J i := {A i g(a j) 0 j l,g G}. A is a strict join of the collection of the sets in J, a contraction to the maximal property. Hence, by taking a finite index subgroup G of G acting trivially on the collection, G is reducible. [S n T] PROPOSITION Suppose that a set G of projective automorphisms in S n (resp. in RP n ) acts on subspaces S 1,...,S l0 and a properly convex domain Ω S n (resp. RP n ), corresponding to independent subspaces V 1,...,V l0 so that V i V j = {0} for i j and V 1 V l0 = R n+1. Let Ω i := Cl(Ω) S i for each i, i = 1,...,l 0. Let λ i (g) denote the largest norm of the eigenvalues of g restricted to V i. We assume that for each S i, G i := {g S i g G} forms a bounded set of automorphisms, and for each S i, there exists a sequence {g i, j G} which has the property { } λi (g i, j ) for each k,k i as j. λ k (g i, j ) Then Cl(Ω) = Ω 1 Ω l0 for Ω j /0, j = 1,...,l 0. PROOF. First, Ω i Cl(Ω) by definition. Since the element of a strict join has a vector that is a linear combination of elements of the vectors in the directions of Ω 1,...,Ω l0, Hence, we obtain Ω 1 Ω l0 Cl(Ω) since Cl(Ω) is convex. Let z = [ v z ] for a vector v z in R n+1. We write v z = v v l0, v j V j for each j, j = 1,...,l 0, which is a unique sum. Then z determines z i = [ v i ] uniquely. Let z be any point. We choose a subsequence of {g i, j } so that {g i, j S i } converges to a projective automorphism g i, : S i S i and λ i, j as j. Then g i, also acts on Ω i. By Proposition 2.20, g i, j (z i ) g i, (z i ) = z i, for a point z i, S i. We also have (2.24) z i = g 1 i, (g i, (z i )) = g 1 i, (lim j g i, j (z i )) = g 1 i, (z i, ).

72 2.5. THE VINBERG DUALITY OF REAL PROJECTIVE ORBIFOLDS 57 Now suppose z Cl(Ω). We have g i, j (z) z i, by the eigenvalue condition. Thus, we obtain z i, Ω i as z i, is the limit of a sequence of orbit points of z. Hence we also obtain z i Ω i by (2.24). We obtain Ω i /0. This also shows that Cl(Ω) = Ω 1 Ω l0 since z {z 1 } {z l0 }. [S n P]For the RP n -version, we lift Ω to an open hemisphere in S n. Then the S n -version implies the RP n -version The Vinberg duality of real projective orbifolds The duality is a natural concept in real projective geometry and it will continue to play an important role in this theory as well The duality. We start from linear duality. Let Γ be a group of linear transformations GL(n + 1,R). Let Γ be the affine dual group defined by {g 1 g Γ}. Suppose that Γ acts on a properly convex cone C in R n+1 with the vertex O. An open convex cone C in R n+1, is dual to an open convex cone C in R n+1 if C R n+1 is the set of linear functionals taking positive values on Cl(C) {O}. C is a cone with the origin as the vertex again. Note (C ) = C, and C must be properly convex since otherwise C o cannot be open. We generalize the notion in Section Now Γ will acts on C. A central dilatational extension Γ of Γ by Z is given by adding a scalar dilatation by a scalar s > 1 for the set R + of positive real numbers. The dual Γ of Γ is a central dilatation extension of Γ. Also, if Γ is cocompact on C if and only if Γ is on C. (See [74] for details.) Given a subgroup Γ in PGL(n + 1,R), a lift in GL(n + 1,R) is any subgroup that maps to Γ bijectively. Given a subgroup Γ in PGL(n + 1,R), the dual group Γ is the image in PGL(n + 1,R) of the dual of any linear lift of Γ. A properly convex open domain Ω in P(R n+1 ) is dual to a properly convex open domain Ω in P(R n+1, ) if Ω corresponds to an open convex cone C and Ω to its dual C. We say that Ω is dual to Ω. We also have (Ω ) = Ω and Ω is properly convex if and only if so is Ω. We call Γ a dividing group if a central dilatational extension acts cocompactly on C with a Hausdorff quotient. Γ is dividing if and only if so is Γ. Define S n := S(R n+1 ). For an open properly convex subset Ω in S n, the dual domain is defined as the quotient in S n of the dual cone of the cone C Ω corresponding to Ω. The dual set Ω is also open and properly convex but the dimension may not change. Again, we have (Ω ) = Ω. If Ω is a domain but not necessarily open, then we define Ω to be the dual domain of Ω o. Given a properly convex domain Ω in S n (resp. RP n ), we define the augmented boundary of Ω (2.25) bd Ag Ω := {(x,h) x bdω,x h, Define the projection h is an oriented sharply supporting hyperspace of Ω} S n S n. Π Ag : bd Ag Ω bdω

73 58 2. PRELIMINARIES by (x,h) x. Each x bdω has at least one oriented sharply supporting hyperspace. An oriented hyperspace is an element of S n since it is represented as a linear functional. Conversely, an element of S n represents an oriented hyperspace in S n. (Clearly, we can do this for RP n and the dual space RP n but we consider only nonoriented supporting hyperspaces.) THEOREM Let A be a subset of bdω. Let A := Π 1 Ag (A) be the subset of bdag (A). Then Π Ag A : A A is a quasi-fibration. PROOF. We take a Euclidean metric on an affine subspace containing Cl(Ω). The sharply supporting hyperspaces at x can be identified with unit normal vectors at x. Each fiber Π 1 Ag (x) is a properly convex compact domain in a sphere of unit vectors through x. We find a continuous unit vector field to bdω by taking the center of mass of each fiber with respect to the Euclidean metric. This gives a local coordinate system on each fiber by giving the origin, and each fiber is a compact convex domain containing the origin. Then the quasi-fibration property is clear now. [S n T] REMARK We notice that for open properly convex domains Ω 1 and Ω 2 in S n (resp. in RP n ) we have (2.26) Ω 1 Ω 2 if and only if Ω 2 Ω 1 REMARK Given a strict join A B for a properly convex compact k-dimensional domain A in RP k and a properly convex compact n k 1-dimensional domain B in RP n k 1, let A denote the dual in RP k of A in RP k and B the dual domain in RP n k 1 of B in RP n k 1. RP k embeds into RP n 1 as P(V 1 ) for the subspace V 1 of linear functionals nullifying vectors in directions of RP n k 1 and RP n k 1 embeds into RP n 1 as P(V 2 ) for the subspace V 2 of linear functionals nullifying the vectors in directions of RP k. Then we have (2.27) (A B) = A B. This follows from the definition and realizing every linear functional as a sum of linear functionals in the direct-sum subspaces. Also, if A S k and B S n k 1 respectively are k-dimensional and (n k 1)-dimensional domains with duals A S k,b S n k 1. We embed S k and S n k 1 to S n 1 as above. Then the above equation also holds. An element (x,h) is bd Ag Ω if and only if x bdω and h is represented by a linear functional α h so that α h ( y) > 0 for all y in the open cone C corresponding to Ω and α h ( v x ) = 0 for a vector v x representing x. Let (x,h) bd Ag Ω. Since the dual cone C consists of all nonzero 1-form α so that α( y) > 0 for all y Cl(C) {O}. Thus, α( v x ) > 0 for all α C and α y ( v x ) = 0. α h C since v x Cl(C) {O}. But h bdω as we can perturb α h so that it is in C. Thus, x is a sharply supporting hyperspace at h bdω. Hence we obtain a continuous map D Ω : bd Ag Ω bd Ag Ω. We define a duality map D Ω : bd Ag Ω bd Ag Ω given by sending (x,h) to (h,x) for each (x,h) bd Ag Ω. PROPOSITION Let Ω and Ω be dual open domains in S n and S n (resp. RP n and RP n ). (i) There is a proper quotient map Π Ag : bd Ag Ω bdω given by sending (x,h) to x.

74 2.5. THE VINBERG DUALITY OF REAL PROJECTIVE ORBIFOLDS 59 (ii) A projective automorphism group Γ acts properly on a properly convex open domain Ω if and only if so Γ acts on Ω (Vinberg s Theorem 2.61 ). (iii) There exists a duality map which is a homeomorphism. D Ω : bd Ag Ω bd Ag Ω (iv) Let A bd Ag Ω be a subspace and A bd Ag Ω be the corresponding dual subspace D Ω (A). A group Γ acts on A so that A/Γ is compact if and only if Γ acts on A and A /Γ is compact. PROOF. We will prove for S n first. (i) Each fiber is a closed set of hyperspaces at a point forming a compact set. The set of sharply supporting hyperspaces at a compact subset of bdω is closed. The closed set of hyperspaces having a point in a compact subset of S n+1 is compact. Thus, Π Ag is proper. Clearly, Π Ag is continuous, and it is an open map since bd Ag Ω is given the subspace topology from S n S n with a product topology where Π Ag extends to a projection. (ii) Straightforward. (See Chapter 6 of [74].) (iii) D Ω has the inverse map D Ω. (iv) The item is clear from (ii) and (iii). [S n T] DEFINITION The two subgroups G 1 of Γ and G 2 of Γ are dual if sending g g 1,T gives us an isomorphism G 1 G 2. A set in A bdω is dual to a set B bdω if D : Π 1 Ag (A) Π 1 Ag (B) is a one-to-one and onto map. REMARK For an open subspace A bdω that is smooth and strictly convex, D induces a well-defined map A bdω A bdω since each point has a unique sharply supporting hyperspace for an open subspace A. The image of the map A is also smooth and strictly convex by Lemma We will simply say that A is the image of D The duality of convex real projective orbifolds with strictly convex boundary. We have O = Ω/Γ for an open properly convex domain Ω, the dual orbifold O = Ω /Γ is a properly convex real projective orbifold. The dual orbifold is well-defined up to projective diffeomorphisms. THEOREM 2.61 (Vinberg). Let O be a strongly tame properly convex real projective open or closed orbifold. The dual orbifold O is diffeomorphic to O. The map given by Vinberg [135] the Vinberg duality diffeomorphism. For an orbifold O with boundary, the map is a diffeomorphism in the interiors O o O o. Also, D O gives us the diffeomorphism O O. (We conjecture that they form a diffeomorphism O O up to isotopies. We also remark that D O D O may not be identity as shown by Vinberg.) Sweeping actions. An action of a projective group G on a properly convex domain Ω is sweeping if Ω/G is compact but not necessarily Hausdorff. A dividing action is sweeping. Recall the commutant H of a group acting on a properly convex domain is the maximal diagonalizable group commuting with the group. (See Vey [134].)

75 60 2. PRELIMINARIES PROPOSITION Suppose that a projective group G acts on an n-dimensional properly convex open domain Ω in S n as a sweeping action. Let L be any subspace where G acts on. Then L Cl(Ω) /0 but L Ω = /0. Suppose that G is semi-simple. Then all the items up to the last one in the conclusion of Proposition 2.49 without discreteness hold. Suppose that G is semi-simple. Replacing the last item, we have: The closure of G in Aut(K) has a virtual center containing a group diagonalizable matrices isomorphic to Z l 0 1 acting trivially on each K i in the conclusion of Proposition G cannot be unipotent. PROOF. Suppose that L Cl(Ω) = /0. Then there is a lower bound to the d-distance from bdω to L. Let x Ω. We denote the space of maximal segments from x and ending in L by L Ω,x. This is a set homeomorphic to S diml. Let l + denote the endpoint of L l ahead of x. Let l Ω,0 denote the end point of l Ω ahead of x, and l Ω,1 denote the end point of l Ω after x. We define a function f : Ω (0, ) given by f (x) = inf{log(l +,l Ω,0,x,l Ω,1 ) l L x }. This is a continuous function. As x bdω, f (x) 0. Since f (g(x)) = f (x) for all x Ω and g G, f induces a continuous map f : Ω/G (0, ). Here, f can take as close value to 0 as one wishes. This contradicts the compactness of Ω/G. Suppose that L Ω /0. Then G acts on the convex domain L Ω open in L. We define a function f : Ω [0, ) given by measuring the Hilbert distance from L Ω. Then f (x) as x bdω L. Again f (g(x)) = f (x) for all x Ω,g G. This induces f : Ω/G [0, ). Since f can take as large value as one wishes for, this contradicts the compactness of Ω/G. Now, we go to the second item. Let G have a G-invariant decomposition R n = V 1 V l0 where G acts irreducibly. The second item follows by Lemma 2 of [134] since any decomposition of R n gives rise to a diagonalizable commutant of rank l 0. For the third item, we prove for the case when G has a G-invariant decomposition R n = V 1 V 2. Then by the second item, G acts on K = K 1 K 2 for properly convex domain K i S(V i ) for i = 1,2. G acts cocompactly on K o and G is a subset of G 1 G 2 R + where G i is isomorphic to G K i extended to act trivially on K i+1 with G i V 2 = I. The closure Ḡ of G is a subgroup of Ḡ 1 Ḡ 2 R + for the closure Ḡ i of G i, i = 1,2 in Aut(K). Ḡ {(g 1,g 2,r) g i Ḡ i,i = 1,2,r R + }. For a fixed pair (g 1,g 2 ), if there are more than one associated r, then we obtain by taking differences that (I,I,r) is in the group for r 1. This implies that Ḡ contains a nontrivial subgroup of R +. Otherwise, Ḡ is in a graph of homomorphism λ : Ḡ 1 Ḡ 2 R +. An orbit of an action of this on the set K1 o Ko 2 (0,1) is in the orbit of the image of λ. Hence, each orbit meets (y 1,y 2 ) (0,1) at a unique point. Thus, we do not have a cocompact action. Furthermore, if we have a G-invariant decomposition K 1 K m, we can use the decomposition K 1 (K 2 K m ). Now, we use the induction, to obtain the result. For the forth item, G acts on invariant subspaces {p} S 1 S 2 S n 2 S n 1. Here, S n 2 must be a supporting hyperspace by the first item. Let A n 1 denote the affine

76 2.5. THE VINBERG DUALITY OF REAL PROJECTIVE ORBIFOLDS 61 subspace bounded by S n 2 containing Ω. Then G restricts to a group of affine transformations of A n 1 acting on Ω in a sweeping manner. Since G is unipotent, G A n 1 is unimodular. By the proof of Lemma 2.5 of [16], G acts transitively on Ω. By a theorem of [101], Ω is a cone. By a theorem of [102], G A n is semi-simple. However, since G is a unipotent group, this is a contradiction. (See Vey [134] also.) PROPOSITION 2.63 (Lemma 1 of Vey [134]). Suppose that a projective group G acts on an (n 1)-dimensional properly convex open domain Ω as a sweeping action. Then the dual group G acts on Ω as a sweeping action also. PROOF. The Vinberg duality map in Theorem 2.61 is from a diffeomorphism Ω Ω. This map is equivariant under the duality homomorphism g g 1 for each g G. Here, G does not need to be a dividing action Extended duality. We can generalize the duality for convex domains as was done in the beginning of Section 2.5. Given a closed convex cone C 1 in R n+1, consider the set of linear functionals in R n+1 taking nonnegative values in C 1. This forms a closed convex cone. We call this a dual cone of C 1 and denote it by C 1. A closed cone C 2 in R n+1, is dual to a closed convex cone C 1 in R n+1 if C 2 is the closure of the set of linear functionals taking nonnegative values in C 1. For a convex compact set U in S n, we form a corresponding convex cone C(U). Then we form C(U) and the image of its projection a convex compact set U in S n. U is properly convex and U o /0 if and only if so is U. Recall the classification of compact convex sets in Proposition 2.8. We denote by S i 0 S n to be a great sphere of dimension i 0. It is a subspace of linear functionals taking zero values in vectors in directions of S n i 0 1 in S n. We can consider it the dual of S i 0 in S n independent of S n i 0 1. A proper-subspace dual K of a properly convex domain K in S i 0 is the dual domain as obtained from considering S i 0 and corresponding vector subspace only. PROPOSITION Let U be a convex compact domain in S n. U is a great i 0 -sphere if and only if U is a great n i 0 1-sphere. If U is a join of a properly convex domain K of dimension j 0 and a great sphere of dimension i 0 with i 0 + j n, then U is a join of S n i 0 j 0 2 and a properly convex domain K in S j 0 of dimension j 0 properly dual to K in S j 0 if i 0 + j < n. U is K S j 0 if i 0 + j = n. If U is a properly convex domain K in a great sphere S i 0 of dimension i 0,i 0 < n, then U is a strict join of the proper-subspace dual K of K in S i 0 and a great sphere S n i 0 1. U is a properly convex domain if and only if so is U. If U is not properly convex and has nonempty interior, then U has empty interior. If U has empty interior, then U is not properly convex and has a nonempty interior provided U is properly convex. In particular if U is an n-hemisphere, then U is a point and vice versa. PROOF. Suppose that U is a great i 0 -sphere. Then C(U) is a subspace of dimension i The set of linear functionals taking 0 values on C(U) form a subspace of dimension n i 0. Hence, U = S(C(U) ) is a great sphere of dimension n i 0 1. The converse is also true.

77 62 2. PRELIMINARIES Suppose that U is not a great sphere. Proposition 2.8 shows us that U is contained in an n-hemisphere. Let S m 0 be the span of U. Here, m 0 = i 0 + j Then U = S j 0 K m 0 j 0 1 for a great sphere S j 0 and a properly convex domain K m 0 j 0 1 in a great sphere of dimension m 0 j 0 1 independent of the first one by Proposition 2.8. C(U) is an open cone in the vector subspace R m 0+1. Then C(U) = R j 0+1 C(K m 0 j 0 1 ) where C(K m 0 j 0 1 ) R m 0 j 0 for independent subspaces R j 0+1 and R m 0 j 0. Let C(U) denote the dual of C(U) in R m 0+1. For f C(U), f = 0 on R j 0+1, and f R m 0 j 0 takes a value 0 in C(K m 0 j 0 1 ). Hence, f : R m 0+1 = R j 0+1 R m 0 j R is in {0} C(K m 0 j 0 1 ). Denote the projection of C(U) in S m 0 by U. Suppose m 0 = n. Then we showed the second case of the second item. Suppose m 0 < n. Then f C(U) is a sum f 1 + f 2 where f 1 is an element of C(U ) extended by setting f 1 R n m 0 {O} = 0 and f 2 is any linear functional satisfying f 2 {O} R m0+1 = 0 where we indicate by {O} the trivial subspaces of the complements. Hence, (( f 2 )) S n m0 1. Hence, U is a strict join of U and a great sphere S n m0 1. The third item is obtained by taking the dual of the second case of the second item. When U is properly convex and open, we can use U o and the closure of its dual. This proves the fourth item. The fifth item is the second case of the second item. K has the empty interior if K is in the first case of the second item or in the third item. The third item corresponds to the case when the dual of K has nonempty interior. This proves the sixth item. The final case is given by the third item where K is a singleton in S Duality and geometric limits. PROPOSITION Suppose that K i be a sequence of properly convex domains in S n geometrically converging to a compact convex domain K with K o /0. Suppose that the dual K has nonempty interior and is not a great sphere. Then K i is a sequence of properly convex domains in S n geometrically converging to the dual K of K. PROOF. Recall the compact metric space of all compact subsets of S n with the Hausdorff metric d H. (See p of Munkres [122].) K i is a Cauchy sequence under the Hausdorff metric d H. By Lemma 11.13, K i is also a Cauchy sequence under the Hausdorff metric of d H of S n. The Hausdorff metric of the space of all compact subsets of S n is a compact metric space. Let K denote the geometric limit of K i. We will show K = K. First, we show K K. Let φ be a limit of a sequence φ i for φ i C(K i ) for each i. By Proposition 2.2, it will be sufficient to show ((φ )) K for every such φ. We may assume that their norms are 1 always. We will show that φ C(K) 0. Let ε 0 := min{π/8,max{d(x, K) x K}} > 0. Choose ε, 0 < ε < ε 0. We claim that K N ε ( K) K i for sufficiently large i. Suppose not. Choose I so that for i > I, d H (K,K i ) < ε. Since K N ε (K i ), any point of K is in ε-neighborhood of K i for i > I. Let x K N ε ( K) K i. Let B ε (x) be an open ball of d-radius ε. Then B ε (x) K. There is a supporting hemisphere H i, not necessarily sharply supporting, so that K i H i and H i meets B ε (x). Take a shortest

78 2.5. THE VINBERG DUALITY OF REAL PROJECTIVE ORBIFOLDS 63 geodesic segment s i from x to H i of d-length < ε. Let ŝ i denote the great circle containing s i. Then s i is perpendicular to H i where it meets. Let y i ŝ i bdb ε (x) to be the point furthest away from H i on ŝ i. Then the d-distance from y i to H i on ŝ i is > ε. This contradicts K N ε (K i ). Since φ i C(Ki ), we have φ i C(K N ε ( K)) 0 for i > I. By letting i, φ C(K N ε ( K)) 0. By letting ε 0, we obtain φ C(K) 0. Hence, φ C(K). We showed K K. Conversely, we show K K. Let S n 1 denote the unit sphere in Rn+1. Let φ C(K). Then φ C(K) S n 1 0. Define ε i = min{φ(c(k i ) S n 1 )}. If ε i 0 for sufficiently large i, then φ C(K i ) for sufficiently large i and ((φ)) K, and we are finished in this case. Suppose ε i < 0 for infinitely many i. By taking subsequence we assume that ε i < 0 for all i. Let H φ, H φ S n, be the hemisphere determined by the nonnegative condition of φ. Then K i H φ /0 for every i. Choose a point y i in K i of maximal distance from H φ. Then d(y i,h φ ) δ i for 0 < δ i π/2. Since K i K, we deduce δ i 0 obviously. Define a distance function d(,h φ ) : S n R +. Then y i is contained in a smooth sphere S δi at the level δ i. Also, K i is contained in the complement of the convex open ball B δi bounded by S δi. Now, we will use come convex affine geometry. Let H i denote the hemisphere whose boundary contains y i and is tangent to S δi and disjoint from B δi. We claim that K i H i : There is an affine subspace A n i of S n containing Hi o H φ containing y i as the origin. Hi o H φ is a region bounded by two parallel hyperspaces in A n i. Let s i denote the shortest segment from y i to H φ. s i is perpendicular to H φ where s i meets it, say x i. Let us use the coordinate system where H i is given by z n = 0 for a coordinate function z n. For any 2-plane P containing s i, P K i is a convex curve with a maximal point y i under d(,h φ ). Hence, P K H φ is disjoint from A n i H i since the derivative function of a convex curve is always strictly decreasing, and at y i the derivative is either zero or the left and right derivatives switch signs. We deduce that the connected set K i H φ is contained in H i. Hence, H i is a sharply supporting hemisphere of K i at y i, and K i H i. Let φ i be the linear functional corresponding to H i. Then φ i C(K i ) 0. The center of H K is on the great circle ŝ i containing s i. The center of H i is on ŝ i and of distance δ i since d(y i,x i ) = δ i. This implies that d(φ,φ i ) = δ i. Hence, φ i φ and we obtain K K by Proposition 2.2.

79

80 CHAPTER 3 Examples of properly convex real projective orbifolds with ends: cusp openings We give examples where our theory applies to. Coxeter orbifolds and the orderability theory for Coxeter orbifolds will be given. Our work jointly done with Gye-Seon Lee and Craig Hodgson generalizing the work of Benoist and Vinberg will be discussed. We state the work of Heusner-Porti on projective deformations of hyperbolic link complement. Also, we state some nice results on finite volume convex real projective structures by Cooper-Long-Tillmann and Crampon-Marquis on horospherical ends and thick and thin decomposition. We also present the nicest cases where a nicest form of the Ehresmann- Thurston-Weil principle applies. For this, we need the principal results from Parts 2 and 3. These are orbifolds admitting complete hyperbolic structures. Finally, we give computations of a specific case with a Mathematica TM file. As stated in the premise, this Chapter will use freely the results of the whole monograph History of examples Originally, Vinberg [136] investigated Coxeter orbifolds as linear groups acting on convex cones. The groundbreaking work produced also many examples of real projective orbifolds and manifolds O suitable to our study. For example, see Kac and Vinberg [91] for a deformation of triangle groups. However, the work was reduced to studying some Cartan form with rank equal to n + 1 for n = dimo. The method turns out to be a bit hard in computing actual examples. Later, Benoist [19] worked out some examples on prisms. Generalizing this, Choi [45] studied the orderability of Coxeter orbifolds after conversing with Kapovich about the deformability. This produced many examples of noncompact orbifolds with convex projective structures which are properly convex by the work of Vinberg. Later Marquis [114] generalized the technique to study the convex real projective structures based on Coxeter orbifolds with truncation polytopes as base spaces. These are compact orbifolds, and so we will not mention these. For compact hyperbolic 3-manifolds, Cooper-Long-Thistlethwaite [57] and [58] produced many examples with deformations using numerical methods. Some of these are exact computations. We now discuss the noncompact strongly tame orbifolds with convex real projective structures. Also, Choi, Hodgson, and Lee [50] computed the deformation spaces of convex real projective structures of some complete hyperbolic Coxeter orbifolds with or without ideal vertices, and Choi and Lee [51] showed that all compact hyperbolic weakly orderable Coxeter orbifolds have the local deformation space of dimension e + 3 where e + is the number of ridges with order 2. These Coxeter orbifolds form a large class of Coxeter orbifolds. 65

81 66 3. EXAMPLES OF PROPERLY CONVEX REAL PROJECTIVE ORBIFOLDS WITH ENDS: CUSP OPENINGS We can generalize these to complete hyperbolic Coxeter orbifolds which are weakly orderable with respect to ideal vertex. Gye-Seon Lee, L. Marquis, and I will prove in later papers related ideal-vertex-orderable Coxeter 3-orbifolds have smooth deformation spaces of computable dimension. For noncompact hyperbolic 3-manifolds, Porti and Tillmann [123]. Cooper-Long- Tillmann [60] and Crampon-Marquis [61] made theories where the ends were restricted to be horospherical. Ballas [4] and [5] made initial studies of deformations of complete hyperbolic 3-manifolds to convex real projective ones. Cooper, Long, and Tillmann [59] have produced a deformation theory for convex real projective manifolds parallel to ours with different types of restrictions on ends such as requiring the end holonomy group to be abelian. They also concentrate on openness of the deformation spaces. We will provide our theory in Part Examples and computations We will give some series of examples due to the author and many other people. Here, we won t give compact examples since we already gave a survey in Choi-Lee-Marquis [53]. Given a polytope P, a face is a codimension-one side of P. A ridge is the codimensiontwo side of P. When P is 3-dimensional, a ridge is called an edge. We will concentrate on n-dimensional orbifolds whose base spaces are homeomorphic to convex Euclidean polyhedrons and whose faces are silvered and each ridge is given an order. For example, a hyperbolic polyhedron with edge angles of form π/m for positive integers m will have a natural orbifold structure like this. DEFINITION 3.1. A Coxeter group Γ is an abstract group defined by a group presentation of form (R i ;(R i R j ) n i j ),i, j I where I is a countable index set, n i j N is symmetric for i, j and n ii = 1. The fundamental group of the orbifold will be a Coxeter group with a presentation R i,i = 1,2,..., f,(r i R j ) n i j = 1 where R i is associated with silvered sides and R i R j has order n i j associated with the edge formed by the intersection of the i-th and j-th sides. Let us consider only the 3-dimensional orbifolds for now. Let P be a fixed convex 3-polyhedron. Let us assign orders at each edge. We let e be the number of edges and e 2 be the numbers of edges of order-two. Let f be the number of sides. For any vertex of P, we will remove the vertex unless the link in P form a spherical Coxeter 2-orbifold of codimension 1. This make P into an 3-dimensional orbifold. Let ˆP denote the differentiable orbifold with sides silvered and the edge orders realized as assigned from P with the above vertices removed. We say that ˆP has a Coxeter orbifold structure. In this chapter, we will exclude a cone-type Coxeter orbifold whose polyhedron has a side F and a vertex v where all other sides are adjacent triangles to F and contains v and all ridge orders of F are 2. Another type we will not study is a product-type Coxeter orbifold whose polyhedron is topologically a polygon times an interval and ridge orders of top and the bottom sides are all 2. These are essentially a lower-dimensional orbifolds. Finally, we will not study Coxeter orbifolds with finite fundamental groups. If ˆP is none of the above type, then ˆP is said to be a normal-type Coxeter orbifold.

82 3.2. EXAMPLES AND COMPUTATIONS 67 A huge class of examples are obtained by complete hyperbolic 3-polytopes with dihedral angles that are submultiples of π. (See Andreev [3] and Roerder [125].) DEFINITION 3.2. The deformation space D( ˆP) of projective structures on an orbifold ˆP is the space of all projective structures on ˆP quotient by isotopy group actions of ˆP. The topology on D( ˆP) is given by as follows: D( ˆP) is a quotient space of the space of the development pairs (dev,h) with topology given by C r -topology defined on compact sets, r 2, for the maps dev : P RP n. We will explain that the space is identical with CDef E ( ˆP) in Proposition Also, CDef E ( ˆP) = CDef E,u,ce ( ˆP) by Corollary A point p of D( ˆP) gives a fundamental polyhedron P in RP 3, well-defined up to projective automorphisms. By Proposition 9.22, D( ˆP) can be identified with CDef E ( ˆP). We concentrate on the space of p D( ˆP) giving a fundamental polyhedron P fixed up to projective automorphisms. This space is called the restricted deformation space of ˆP and denoted by D P ( P). ˆ A point t in D P ( P) ˆ is said to be hyperbolic if it is given by a hyperbolic structure on ˆP. A point p of D( ˆP) always determines a fundamental polyhedron P up to projective automorphisms because p determines the holonomy up to conjugations and hence reflections corresponding to sides up to conjugations also. We wish to understand the space where the fundamental polyhedron is always projectively equivalent to P. We call this the restricted deformation space of ˆP and denote it by D P ( ˆP). The work of Vinberg [136] implies that each element of D P ( ˆP) gives a convex projective structure (see Theorem 2 of [45]). That is, the image of the developing map of the orbifold universal cover of ˆP is projectively isomorphic to a convex domain in RP 3 and the holonomy is a discrete faithful representation. Now, we state the key property in this paper: DEFINITION 3.3. We say that P is orderable if we can order the sides of P so that each side meets sides of higher index in less than or equal to 3 edges. A pyramid with a complete hyperbolic structure and dihedral angles that are submultiples of π. is obvious example. See Proposition 4 of [45] worked out with J. R. Kim. An example is a drum-shaped convex polyhedron which has top and bottom sides of same polygonal type and each vertex of the bottom side is connected to two vertices in the top side and vice versa. Another example will be a convex polyhedron where each pair of the interiors of nontriangular sides are separated by a union of triangles. In these examples, since nontriangular sides are all separated by the union of triangular sides, the sides are either level 0 or level 1 and hence they satisfy the trivalent condition. A dodecahedron would not satisfy the condition. If P is compact, then Marquis [114] showed that P is a truncation polytope; that is, one starts from a tetrahedron and cut a neighborhood of a vertex so as to change the combinatorial type near that vertex only. Many of these can be realized as a complete hyperbolic polytope with dihedral angle submultiples of π. If P is not compact, we don t have the classification. Also, infinitely many of these can be realized as a complete hyperbolic polytope with dihedral angles that are submultiples of π. (D. Choudhury was first to show this.) DEFINITION 3.4. We denote by k(p) the dimension of the projective group acting on a convex polyhedron P. The dimension k(p) of the subgroup of G acting on P equals 3 if P is a tetrahedron and equals 1 if P is a cone with base a convex polyhedron which is not a triangle. Otherwise, k(p) = 0.

83 68 3. EXAMPLES OF PROPERLY CONVEX REAL PROJECTIVE ORBIFOLDS WITH ENDS: CUSP OPENINGS THEOREM 3.5. Let P be a convex polyhedron and ˆP be given a normal-type Coxeter orbifold structure. Let k(p) be the dimension of the group of projective automorphisms acting on P. Suppose that ˆP is orderable. Then the restricted deformation space of projective structures on the orbifold ˆP is a smooth manifold of dimension 3 f e e 2 k(p) if it is not empty. If we start from a complete hyperbolic polytopes with dihedral angles that are submultiples of π, we know that the restricted deformation space is not empty. If we assume that P is compact, then we refer to Marquis [114] for the complete theory. The topic is not within the scope of this monograph. DEFINITION 3.6. Let P be a 3-dimensional hyperbolic Coxeter polyhedron, and let ˆP denote its Coxeter orbifold structure. Suppose that t is the corresponding hyperbolic point of D P ( ˆP). We call a neighborhood of t in D P ( ˆP) the local restricted deformation space of P. We say that ˆP is projectively deformable relative to the mirrors, or simply deforms rel mirrors, if the dimension of its local restricted deformation space is positive. Conversely, we say that ˆP is projectively rigid relative to the mirrors, or rigid rel mirrors, if the dimension of its local restricted deformation space is 0. The following theorem describes the local restricted deformation space for a class of Coxeter orbifolds arising from ideal hyperbolic polyhedra, i.e. polyhedra with all vertices on the sphere at infinity. THEOREM 3.7 (Choi-Hodgson-Lee [50]). Let P be an ideal 3-dimensional hyperbolic polyhedron whose dihedral angles are all equal to π/3, and suppose that ˆP is given its Coxeter orbifold structure. If P is not a tetrahedron, then a neighborhood of the hyperbolic point in D P ( P) ˆ is a smooth 6-dimensional manifold. The main ideas in the proof of Theorem 3.7 are as follows. We first show that D P ( ˆ P) is isomorphic to the solution set of a system of polynomial equations following ideas of Vinberg [136] and Choi [45]. Since the faces of P are fixed, each projective reflection in a face of the polyhedron is determined by a reflection vector b i. We then compute the Jacobian matrix of the equations for the b i at the hyperbolic point. This reveals that the matrix has exactly the same rank as the Jacobian matrix of the equations for the Lorentzian unit normals of a hyperbolic polyhedron with the given dihedral angles. By the infinitesimal rigidity of the hyperbolic structure on ˆP, this matrix is of full rank and has kernel of dimension six; the result then follows from the implicit function theorem. In fact, we can interpret the infinitesimal projective deformations as applying infinitesimal hyperbolic isometries to the reflection vectors. We can generalize the above theorem slightly as we discussed. DEFINITION 3.8. Given a hyperbolic n-orbifold X with totally geodesic boundary component diffeomorphic to an (n 1)-orbifold Σ. Let X denote the universal cover in the Klein model B in S n. Let Γ be the group of deck transformations considered as projective automorphisms of S n. Then a complete hyperbolic hyperspace Σ covers Σ. Every component of the inverse image of Σ is of form g( Σ) for g π 1 (X). A point v Σ Sn B A (B) is projectively dual to the hyperspace containing Σ with respect to the bilinear form B. (See Section ) Then we form the join C := v Σ Σ {v Σ }. Then we form Ĉ := X g Γ g(c). Ĉ/Γ is an n-orbifold with radial ends. We call the ends the hyperideal ends.

84 3.2. EXAMPLES AND COMPUTATIONS 69 A point of D P ( ˆ P) corresponding to a hyperbolic n-orbifold with hyperideal ends added will be called a hyperbolic point again. An ideal 3-dimensional hyperbolic polyhedron with possibly hyperideal vertices is a compact convex polyhedron with vertices outside B removed. We will generalize this further in Section COROLLARY 3.9 (Choi-Hodgson-Lee). Let P be an ideal 3-dimensional hyperbolic polyhedron with possibly hyperideal vertices whose dihedral angles are of form π/p for integers p 3, and suppose that ˆP is given its Coxeter orbifold structure. If P is not a tetrahedron, then a neighborhood of the hyperbolic point in D P ( P) ˆ is a smooth 6-dimensional manifold. We did not give proof for the case when some edges orders are greater than equal to 4 in the article [50]. However, the same proof will apply as first observed by Hodgson: Recall that (p, q, r)-triangle reflection orbifold have rigid hyperbolic structure and rigid representations in the isometry group of hyperbolic space. Horospherical (3,3,3)-triangle reflection orbifolds of the end can be replaced with totally geodesic (p, q, r)-triangle reflection orbifold for p, q, r 3 by generalizing the Garland-Raghunathan-Weil rigidity [71] and [137] with Theorem 7 (Sullivan rigidity) of [129]. These examples are convex by the work of Vinberg [136]. Corollary 3.25 implies the proper convexity Vertex orderable Coxeter orbifolds Vinberg theory. Let ˆP be a Coxeter orbifold of dimension n. Let P be the fundamental convex polytope of ˆP. The reflection is given by a point, caller a reflection point, and a line. Let R i be a projective reflection on a hyperspace S i containing a side of P. Then we can write R i := I α i v i where α i is zero on S i and v i is the reflection vector and α i ( v i ) = 2. Given a reflection group Γ. We form a Cartan matrix A(Γ) given by a i j := α i ( v j ). Vinberg [136] proved that the following conditions are necessary and sufficient for Γ to be a linear Coxeter group: (C1) a i j 0 for i j, and a i j = 0 if and only if a ji = 0. (C2) a ii = 2;and (C3) for i j, a i j a ji 4 or a i j a ji = 4cos 2 ( π n i j ) an integer n i j. The Cartan matrix is a f f -matrix when P has f sides. Also, a i j = a ji for all i, j if Γ is conjugate to a reflection group in O + (1,n). This condition is the condition of ˆP to be a hyperbolic Coxeter orbifold. The Cartan matrix is determined only up to an action of the group D f, f of nonsingular diagonal matrices: This is due to the ambiguity of choices A(Γ) DA(Γ)D 1 for D D f, f. α i c i α i, v i 1 c i v i,c i > 0. Hence, the set of all cyclic invariants of form a i1 i 2 a i2 i 3 a ir i 1 classifies the linear Coxeter group up to the conjugation.

85 70 3. EXAMPLES OF PROPERLY CONVEX REAL PROJECTIVE ORBIFOLDS WITH ENDS: CUSP OPENINGS The classification of convex real projective structures on triangular reflection orbifolds: We will follow Kac-Vinberg [91]. Let ˆT be a 2-dimensional Coxeter orbifold based on a triangle T. Let the edges of T be silvered. Let the vertices be given orders p,q,r where 1/p + 1/q + 1/r 1. If 1/p + 1/q + 1/r 1, then the universal cover T of ˆT is a properly convex domain or a complete affine plane by Vinberg [136]. We can find the topology of D( ˆT ) as Goldman did in his senior thesis [73]. We may put T as a standard triangle with vertices e 1 := [1,0,0], e 2 := [0,1,0], e 3 := [0,0,1]. Let R i be the reflection on a line containing [ e i 1 ],[ e i+1 ] and with a reflection vertex [ v i ]. Let α i denote the linear function on R 3 taking zero values on e i 1 and e i+1. We choose v i to satisfy α i ( v i ) = 1. When 1/p + 1/q + 1/r = 1, the triangular orbifold admits a compatible Euclidean structure. When 1/p+1/q+1/r < 1, the triangular orbifold admits a hyperbolic structure not necessarily compatible with the real projective structure. A linear Coxeter group Γ is hyperbolic if and only if the Cartan matrix A of Γ is indecomposable, of negative type, and equivalent to a symmetric matrix of signature (1, n). Assume that no p,q,r is 2 and 1/p + 1/q + 1/r < 1. Let a i j denote the entries of the Cartan matrix. It satisfies a 12 a 21 = 4cos 2 π/p,a 23 a 32 = 4cos 2 π/q,a 13 a 31 = 4cos 2 π/r. There are only two cyclic invariants a 12 a 23 a 31 and a 13 a 32 a 21 satisfying a 12 a 23 a 31 a 13 a 32 a 21 = 64cos 2 π/pcos 2 π/qcos 2 π/r. Then the triple invariant a 12 a 23 a 31 R + classifies the conjugacy classes of Γ. A point will correspond to a hyperbolic structure. For different points, they are properly convex. Since a i j = a ji for geometric cases, we obtain that a 12 a 23 a 31 = 2 3 cos(π/p)cos(π/q)cos(π/r) gives the unique hyperbolic points. We define for this orbifold D( ˆT ) := R + the space of the triple invariants. Unique point is Euclidean or hyperbolic. EXAMPLE 3.10 (Lee s example). Consider the Coxeter orbifold ˆP with the underlying space on a polyhedron P with the combinatorics of a cube with all sides mirrored and all edges given order 3 but vertices removed. By the Mostow-Prasad rigidity and the Andreev theorem, the orbifold has a unique complete hyperbolic structure. There exists a six-dimensional space of real projective structures on it by Theorem 3.7 where one has a projectively fixed fundamental domain in the universal cover. There are eight ideal vertices of P corresponding to eight ends of ˆP. Each end orbifold is a 2-orbifold based on a triangle with edges mirrored and vertex orders are all 3. Each end orbifold has a real projective structure and hence is characterized by the triple invariant. Thus, each end has a neighborhood diffeomorphic to the 2-orbifold multiplied by (0, 1). The eight triple invariants are related when we are working on the restricted deformation space since the deformation space is only six dimensional. (They might be independent in the full deformation space as M. Davis and R. Green observed. ) The end mappings. We will give some explicit class of examples where we can control the end structures. I worked this out with Greene, Gye-Seon Lee, and Marquis starting from the workshop at the ICERM in DEFINITION Let P be a simple 3-polytope given a natural number 2 on each edge. Let V be a set of ideal vertices chosen ahead so that P V has an orbifold structure

86 3.3. THE WORK OF OTHER GROUPS 71 ˆP. Let the faces of ˆP be given an ordering. Each face has at most one vertex in V. Each vertex in V has three edges ending there and the edges have order 3 only. For each face F i, let N i denote the number of edges of order 2 or in the faces of higher ordering. Then ˆP is V -orderable if N i is 1 for any face F i containing a vertex in V and N i 3 for face F i containing no vertex in V. Let V O denote the disjoint union of end orbifolds corresponding to the set of ideal vertices V. THEOREM 3.12 (Choi-Greene-Lee-Marquis [49]). Suppose that P with a set of vertices V is V -orderable. Suppose P admits a Coxeter orbifold structure with a convex real projective structure. Then the function D(O) D( V O) is onto. We also make a generalization to weakly V -orderable orbifolds, where we let N i be redefined as the the number of edges of order 2 and in the faces of higher ordering. THEOREM 3.13 (Choi-Greene-Lee-Marquis [49]). Suppose that P with a set of vertices V is weakly V -orderable. Suppose P admits a Coxeter orbifold structure with a complete hyperbolic structure. Then the function D(O) D( V O) is locally surjective at the hyperbolic point The work of other groups For closed hyperbolic manifolds, the deformation spaces of convex structures on manifolds were extensively studied by Cooper-Long-Thistlethwaite [57] and [58] The work of Heusener-Porti. DEFINITION Let N be a closed hyperbolic manifold of dimension equal to 3. We consider the holonomy representation of N ρ : π 1 (N) PSO(3,1) PGL(4,R). A closed hyperbolic three manifold N is called infinitesimally projectively rigid if H 1 (π 1 (N),sl(4,R) Adρ ) = 0. DEFINITION Let M denote a compact three-manifold with boundary a union of tori and whose interior is hyperbolic with finite volume. M is called infinitesimally projectively rigid relative to the cusps if the inclusion M M induces an injective homomorphism H 1 (π 1 (M),sl(4,R) Adρ ) H 1 ( M,sl(4,R) Adρ ). THEOREM 3.16 (Heusener-Porti [86]). Let M be an orientable 3-manifold whose interior has a complete hyperbolic metric with finite volume. If M is infinitesimally projectively rigid relative to the cusps, then infinitely many Dehn fillings on M are infinitesimally projectively rigid. THEOREM 3.17 (Heusener-Porti [86]). Let M be an orientable 3-manifold whose interior has a complete hyperbolic metric of finite volume. If a hyperbolic Dehn filling N on M satisfies: (i) N is infinitesimally projectively rigid, (ii) the Dehn filling slope of N is contained in the (connected) hyperbolic Dehn filling space of M,

87 72 3. EXAMPLES OF PROPERLY CONVEX REAL PROJECTIVE ORBIFOLDS WITH ENDS: CUSP OPENINGS then infinitely many Dehn fillings on M are infinitesimally projectively rigid. The complete hyperbolic manifold M that is the complement of a figure-eight knot in S 3 is infinitesimally projectively rigid. The infinitely many Dehn fillings on M is infinitesimally projectively rigid. They also showed the following: For a sufficiently large positive integer k, the homology sphere obtained by 1 k Dehn filling on the figure eight knot is infinitesimally not projectively rigid. Since the Fibonacci manifold M k is a branched cover of S 3 over the figure eight knot complements, for any k N, the Fibonacci manifold M k is not projectively rigid. They also showed: All but finitely many punctured torus bundles with tunnel number one are infinitesimally projectively rigid relative to the cusps. All but finitely many twist knots complements are infinitesimally projectively rigid relative to the cusps Ballas s work on ends. The following are work of Ballas [4] and [5]. Let M be the complement in S 3 of 4 1 (the figure-eight knot), 5 2, 6 1, or (the Whitehead link). Then M does not admit strictly convex deformations of its complete hyperbolic structure. Let M be the complement of a hyperbolic amphichiral knot, and suppose that M is infinitesimally projectively rigid relative to the boundary and the longitude is a rigid slope. Then for sufficiently large n, there is a one-dimensional family of strictly convex deformations of the complete hyperbolic structure on M(n/0). Let M be the complement in S 3 of the figure-eight knot. There exists ε such that for each s (ε,ε), ρ s is the holonomy of a finite volume properly convex projective structure on M. Furthermore, when s 0, this structure is not strictly convex Finite volume strictly convex real projective orbifolds with ends. This was studied by two independent groups. We summarize their main results. The Hilbert metric is a complete Finsler metric on a properly convex set Ω. This is the hyperbolic metric in the Klein model when Ω projectively diffeomorphic to a standard ball. A simplex with the Hilbert metric is isometric to a normed vector space, and appears in a natural geometry on the Lie algebra sl(n,r). A singular version of this metric arises in the study of certain limits of projective structures. The Hilbert metric has a Hausdorff measure and hence a notion of finite volume. THEOREM 3.18 (Choi [39], Cooper-Long-Tillmann [60], Crampon-Marquis [61]). For each dimension n 2 there is a Margulis constant µ n > 0 with the following property. If M is a properly convex projective n-manifold and x is a point in M, then the subgroup of π 1 (M,x) generated by loops based at x of length less than µ n is virtually nilpotent. In fact, there is a nilpotent subgroup of index bounded above by m = m(n). Furthermore, if M is strictly convex and finite volume, this nilpotent subgroup is abelian. If M is strictly convex and closed, this nilpotent subgroup is trivial or infinite cyclic. THEOREM 3.19 (Cooper-Long-Tillmann [60], Crampon-Marquis [61]). Each end of a strictly convex projective manifold or orbifold of finite volume is horospherical. THEOREM 3.20 ((Relatively hyperbolic). Cooper-Long-Tillmann [60], Crampon Marquis [61]). Suppose M = Ω/Γ is a properly convex manifold of finite volume which is the interior of a compact manifold N and the holonomy of each component of N is topologically parabolic. Then the following are equivalent:

88 3.4. NICEST CASES 73 1 Ω is strictly convex, 2 Ω is C 1, 3 π 1 (N) is hyperbolic relative to the subgroups of the boundary components. Here, the definition of the term topologically parabolic is according to [60]. This is not a Lie group definition but a topological definition. We have found a generalization Theorem and its converse Theorem in Chapter Nicest cases We will now present the cases when the theory presented in this monograph works best. DEFINITION A countable group G satisfies the property (NS) if every normal solvable subgroup of a finite-index normal subgroup G is central in G. By Corollary 2.50, the fundamental group of a closed orbifold admitting a properly convex structure has the property (NS). Clearly a virtually abelian group satisfies (NS). EXAMPLE Let M be a complete hyperbolic 3-orbifold and each end orbifold has a sphere or a disk as the base space. The end fundamental group is generated by a finite order elements. By Lemma 3.23, we obtain that properly convex real projective structure on M have lens-shaped or horospherical radial ends only. We need the end classification results from Chapters 4, 6, and 8 to prove the following. Let g π 1 (O). Using the choice of representing matrix of g as in Remark 1.1, we let λ x (g) denote the eigenvalue of holonomy of g associated with the vector in direction of x if x is a fixed point of g. The holonomy group of π 1 (O) can be lifted to SL ± (n + 1,R) so that λ vẽ (g) = 1 for the holonomy of every g π 1 (Ẽ) where vẽ is a p-end vertex of a p-end Ẽ corresponding to E. Then we say that E or Ẽ satisfies the unit middle eigenvalue condition with respect to vẽ or the R-p-end structure. Suppose that E is a T -end. If the hyperspace containing the ideal boundary component SẼ of p-end Ẽ of E corresponds to 1 as the eigenvalue of the dual of the holonomy of every g π 1 (Ẽ), then we say we say that E or Ẽ satisfies the unit middle eigenvalue condition with respect to SẼ or the T-p-end structure. LEMMA Suppose that O is a strongly tame properly convex real projective orbifold with radial ends. Assume that the end fundamental group π 1 (E) of an end E satisfies (NS). Let E be an R-end or a T -end. Suppose one of the following holds: π 1 (E) is virtually generated by finite order elements or is simple, or the end holonomy group of E satisfies the unit middle eigenvalue condition. Then the following hold: the end E is either properly convex generalized lens-shaped R-end or a lensshaped T-end or is horospherical. If the end E furthermore has a virtually abelian end holonomy group, then E is of lens-shaped R-end or of lens-shaped T-end or is a horospherical end. PROOF. We suppose that O is a convex domain in of S n. First, let E be an R-end. The map g ΓẼ λ vẽ (g) R +

89 74 3. EXAMPLES OF PROPERLY CONVEX REAL PROJECTIVE ORBIFOLDS WITH ENDS: CUSP OPENINGS is a homomorphism. Thus, λ vẽ (g) = 1 for g ΓẼ since the end holonomy group is simple or virtually generated by the finite order elements. Each R-end is either complete, properly convex, or is convex but not properly convex and not complete by Section 4.1. Suppose that Ẽ is complete. Then Theorem 4.8 shows that either Ẽ is horospherical or each element g, g π 1 (Ẽ) has at most two norms of eigenvalues where two norms for an element are realized. Since the multiplications of all eigenvalues equals 1, we obtain λ n+1 r 1 (g)λ vẽ (g) r = 1 for some integer r,1 r n and the other norm λ 1 (g) of the eigenvalues. The second case cannot happen. Suppose that Ẽ is properly convex. Then the uniform middle eigenvalue condition holds by Remark 6.14 since λ vẽ (g) = 1 for all g. (See Definition 6.4.) By Theorem 6.7, Ẽ is of generalized lens-type. Finally, Corollary 8.45 rules out the case when Ẽ is convex but not properly convex and not complete. Now, let E be a T-end. By dualizing the above, E satisfies the uniform middle eigenvalue condition (see Definition 6.5). Theorem 6.37 implies the result. [SS n ] This is Theorem A of Marquis [117] when the orbifold is a Coxeter one. THEOREM Suppose that O is a strongly tame properly convex real projective orbifold with R-ends or T-ends. Suppose that each end fundamental group satisfies property (NS) and is virtually generated by finite order elements, or is simple or satisfies the unit middle eigenvalue condition. Then the holonomy is in Hom s E,u,ce (π 1(O),PGL(n + 1,R)). PROOF. Suppose that E is an R-end. Let Ẽ be a p-end corresponding to E and vẽ be the p-end vertex. By Lemma 3.23, we obtain the R-end is lens-type or horospherical. We prove the uniqueness of the fixed point under h(π 1 (Ẽ)): Suppose that x is another fixed point of h(π 1 (Ẽ)). Since π 1 (Ẽ) is as in the premise, the eigenvalue λ x (g) for every g π 1 (Ẽ) associated with x is always 1. In the horospherical case x = vẽ since the cocompact lattice action on a cusp group fixes a unique point in RP n. Now consider the lens case. The uniform middle eigenvalue condition with respect to vẽ and x holds by Remark 6.14 since λ x (g) = 1 for all g. In the lens case, we see from the proof of Proposition that x is on a hyperspace S disjoint from vẽ and h(π 1 (Ẽ)) cannot satisfy the uniform middle eigenvalue condition with respect to x; or else x,vẽ is on a line fixed by h(π 1 (Ẽ)) and h(π 1 (Ẽ)) cannot satisfy the uniform middle eigenvalue condition with respect to vẽ. Lemma 3.23 completes the proof. Suppose that E is a T -end. The proof of Proposition 6.33 shows that the hyperspace containing SẼ corresponds to vẽ for the R-p-end Ẽ corresponding to the dual of the T-pend Ẽ and vice verse. Hence, the result follows from the R-end part of the proof. Theorems 3.24, 1.29, and 1.21 imply the following: COROLLARY Let O be a noncompact strongly tame SPC n-dimensional orbifold with R-ends and T-ends and satisfies (IE) and (NA). Suppose that each end fundamental group is generated by finite order elements or is simple. Suppose each end fundamental group satisfies (NS). Assume O = /0, and that the nilpotent normal subgroups of every

90 3.5. TWO SPECIFIC EXAMPLES 75 finite-index subgroup of π 1 (O) are trivial. Then CDef E (O) = CDef E,u,ce (O) and hol maps the deformation space CDef E (O) of SPC-structures on O homeomorphic to a union of components of which is a union of components of rep s E,u,ce (π 1(O),PGL(n + 1,R)) rep E,u,ce (π 1 (O),PGL(n + 1,R)) and rep E (π 1 (O),PGL(n + 1,R)). The same can be said for SDef E (O) = SDef E,u,ce (O). These type of deformations from structures with cusps to ones with lens-shaped ends are realized in our main examples as stated in Section 3.2. We need the restrictions on the target space since the convexity of O is not preserved under the hyperbolic Dehn surgery deformations of Thurston, as pointed out by Cooper at ICERM in September Since finite-volume hyperbolic n-orbifolds satisfy (IE) and (NA) (see P.151 of [113] for example), strongly tame properly convex orbifolds admitting complete hyperbolic structures end fundamental groups generated by finite order elements will satisfy the premise. Hence, 2h 1 1 and the double of the simplex orbifold do also. Since Coxeter orbifolds satisfy the above properties, we obtain a simple case: COROLLARY Let O be a strongly tame Coxeter n-dimensional orbifold, n 3, with only R-ends admitting a finite-volume complete hyperbolic structure. Then SDef E,u,ce (O) is homeomorphic to a union of components of which is a union of components of Finally, rep s E,u,ce (π 1(O),PGL(n + 1,R)) rep E (π 1 (O),PGL(n + 1,R)) SDef E,u,ce (O) = SDef E (O) Two specific examples The example of S. Tillmann is an orbifold on a 3-sphere with singularity consisting of two unknotted circles linking each other only once under a projection to a 2-plane and a segment connecting the circles (looking like a linked handcuff) with vertices removed and all arcs given as local groups the cyclic groups of order three. (See Figure 3.) This is one of the simplest hyperbolic orbifolds in Heard, Hodgson, Martelli, and Petronio [85] labeled 2h 1 1. The orbifold admits a complete hyperbolic structure since we can start from a complete hyperbolic tetrahedron with four dihedral angles equal to π/6 and two equal to 2π/3 at a pair of opposite edge e 1 and e 2. Then we glue two faces adjacent to e i by an isometry fixing e i for i = 1,2. The end orbifolds are two 2-spheres with three cone points of orders equal to 3 respectively. These end orbifolds always have induced convex real projective structures in dimension 2, and real projective structures on them have to be convex. Each of these is either the quotient of a properly convex open triangle or a complete affine plane as we saw in Lemma 3.23.

91 76 3. EXAMPLES OF PROPERLY CONVEX REAL PROJECTIVE ORBIFOLDS WITH ENDS: CUSP OPENINGS FIGURE 1. A convex developing image example of a tetrahedral orbifold of orders 3,3,3,3,3,3. Porti and Tillmann [123] found a two-dimensional solution set from the complete hyperbolic structure by explicit computations. His main questions are the preservation of convexity and realizability as convex real projective structures on the orbifold. The another main example can be obtained by doubling a complete hyperbolic Coxeter orbifold based on a convex polytopes. We take a double D T of the reflection orbifold based on a convex tetrahedron with orders all equal to 3. This also admits a complete hyperbolic structure since we can take the two tetrahedra to be the regular complete hyperbolic tetrahedra and glue them by hyperbolic isometries. The end orbifolds are four 2-spheres with three singular points of orders 3. Topologically, this is a 3-sphere with four points removed and six edges connecting them all given order 3 cyclic groups as local groups. THEOREM Let O denote the hyperbolic 3-orbifold D T. We assign the R-type to each end. Then SDef E (O) equals SDef E,u,ce (O) and hol maps SDef E (O) as an onto-map to a component of characters rep E (π 1 (O),PGL(4,R)) containing a hyperbolic representation which is also a component of rep E,u,ce (π 1 (O),PGL(4,R)). In this case, the component is a cell of dimension 4. PROOF. We prove for S n. The end orbifolds have Euler characteristics equal to zero and all the singularities are of order 3. By Corollary 3.25, SDef E (O) equals SDef E,u,ce (O). Each of the ends has to be either horospherical or lens-shaped or totally geodesic radial type. Let E O denote the union of end orbifolds of O. In [42], we showed that the triangulated real projective structures on the ends determined the real projective structure on O. First, there is a map SDef E (O) CDef( E O)

92 3.5. TWO SPECIFIC EXAMPLES 77 given by sending the real projective structures on O to the real projective structures of the ends. (Here if E O has many components, then CDef( E O) is the product space of the deformation space of all components.) Let J be the image. Let µ be an element of SDef E (O). The universal cover O is a properly convex domain in S 3. Each singular geodesic arc in O connects one of the p-end vertices to the another. The developing image of O is a convex open domain and the developing map is a diffeomorphism. The developing images of singular geodesic arcs form geodesics meeting at vertices transversally. There exists two convex tetrahedra T 1 and T 2 with vertices removed in O. They are adjacent and their images under π 1 (O) tessellate O. The end orbifold is so that if given an element of the deformation space, then the geodesic triangulation is uniquely obtained. Hence, there is a proper map from SDef E (O) to the space of invariants of the triangulations as in [42], i.e, the product space of crossratios and Goldman-invariant spaces. Now O is the orbifold obtained from doubling a tetrahedron with edge orders 3,3,3. We consider an element of SDef E (O). Since it is convex, we triangulate O into two tetrahedra, and this gives a triangulation for each end orbifold diffeomorphic to S (i) 3,3,3, i = 1,2,3,4, corresponding to four ends, each of which gives us triangulations into two triangles. We can derive from the result of Goldman [76] and Choi-Goldman [48] that given projective invariants ρ (i) 1,ρ(i) 2,ρ(i) 2,σ(i) 1,σ(i) ρ (i) 1 ρ(i) 2 ρ(i) 3 = σ (i) 1 σ (i) 2 2 for each of the two triangles satisfying, we can determine the structure on S(i) for i = 1,2,3,4 completely. For S (i) 3,3,3 with a convex real projective structure and divided into two geodesic triangles, we compute these invariants ρ (i) 1,ρ(i) 2,ρ(i) 2,σ(i) 1,σ(i) 2 for one of the triangles 3,3,3 (3.1) s 2 i + sτ + 1,s2 i + s iτ + 1,s 2 i + s iτ + 1, t i ( s 2 i + s i τ + 1 ), 1 t i ( s 2 i + s i τ + 1 )( s 2 i + s iτ + 1 ) and for the other triangle the corresponding invariants are (3.2) 1 (s 2 s 2 i + s iτ + 1), i t i ( s s 2 3 i + s i τ + 1 ), i 1 s 2 (s 2 i + s i τ + 1), 1 i s 2 (s 2 i + s i τ + 1), i 1 ( s 2 s 3 i t i + s i τ + 1 )( s 2 i + s i τ + 1 ) i where s i,t i, i = 1,2,3,4, are Goldman parameters and τ = 2cos2π/3. (See [33].) Since E O is a disjoint union of four spheres with singularities (3,3,3), CDef( E O) is parameterized by s i,t i and hence is a cell of dimension 8. (This can be proved similarly to [54].) The set J is given by projective invariants of the (3,3,3) boundary orbifolds satisfying equations from the facts that the cross ratio of an edge are same from one boundary orbifold to the other and that the products of Goldman σ-invariants for each tetrahedron equal 1. By the method of [42] developed by the author, we obtain the equations that J satisfies.

93 78 3. EXAMPLES OF PROPERLY CONVEX REAL PROJECTIVE ORBIFOLDS WITH ENDS: CUSP OPENINGS These are t 1 t 2 t 3 t s 2 i i=1 t i i=1 s 3 i The equation is solvable: s 2 i + s i τ + 1 = s 2 j + s j τ + 1,i, j = 1,2,3,4 (s 2 i + s i τ + 1) = 1 s 2 (s 2 j + s i τ + 1),i, j = 1,2,3,4 j ( s 2 i + s i τ + 1 ) = 1 t 1 t 2 t 3 t 4 ( s 2 i + s i τ + 1 ) = 4 i=1 1 s 3 i t i 4 i=1 ( s 2 i + s i τ + 1 ) 2 ( s 2 i + s i τ + 1 ). s 1 = s 2 = s 3 = s 4 = s,t 1 t 2 t 3 t 4 = C(s) for a constant C(s) > 0 depending ons. Thus J is contained in the solution subspace C, a 4-dimensional cell in CDef( E O). Conversely, given an element of C, we can assign invariants at each edge of the tetrahedron and the Goldman σ-invariants at the vertices if the invariants satisfy the equations. This is given by starting from the first convex tetrahedron and gluing one by one using the projective invariants (see [42] and [36]): Let the first one by always be the standard tetrahedron with vertices [1,0,0,0],[0,1,0,0],[0,0,1,0], and [0,0,0,1] and we let T 2 a fixed adjacent tetrahedron with vertices [1,0,0,0],[0,1,0,0],[0,0,1,0] and [2,2,2, 1]. Then projective invariants will determine all other tetrahedron triangulating O. Given any deck transformation γ, T 1 and γ(t 1 ) will be connected by a sequence of tetrahedrons related by adjacency and their pasting maps are completely determined by the projective invariants, where cross-ratios do not equal 0. Therefore, as long as the projective invariants are bounded, the holonomy transformations of the generators are bounded. Corollary 3.26 shows that these corresponds to elements of SDef E,u,ce (O) = SDef E (O). (This method was spoken about in our talk in Melbourne, May 18, 2009 [36].) Hence, we showed that SDef E,u,ce (O) is parameterized by the solution set C. Thus J = C since each element of C gives us an element of SDef E (O). [SS n ] The dimension is one higher than that of the deformation space of the reflection 3- orbifold based on the tetrahedron. Thus we have examples not arising from reflection ones here as well. See the Mathematica files [34] for a different explicit method of solutions. Also, see [35] to see how to draw Figure 1. We remark that the above theorem can be generalized to orders 3 with hyperideal ends with similar computations. See [34] for examples to modify orders and so on.

94 Part 2 The classification of radial and totally geodesic ends.

95 The purpose of this part is to understand the structures of ends of real projective n- dimensional orbifolds for n 2. In particular, we consider the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyperideal ends gives us examples. For this, we will study the natural conditions on eigenvalues of holonomy representations of ends when these ends are manageably understandable. This is the most technical part of the monograph containing large number of results useful in other two parts. We begin the study of radial ends in Chapter 4. We will divide radial ends into complete affine radial ends, properly convex ends, and convex but not complete affine ends. We give some examples of radial ends. We will also classify the complete affine ends in this chapter. In Chapter 5, we study the affine actions dual to the R-end theory. This is the major technical section in this part. We consider the case when there is a discrete affine action of a group Γ acting cocompactly on a properly convex domain Ω in the boundary of the affine subspace A n in RP n or S n. We study the convex domain U in an affine space A n with the boundary in Ω. The domain U have asymptotic hyperspaces at each point of bdω if and only if Γ satisfies the uniform middle eigenvalue condition with respect to bda n. The idea is to study the flow on the affine bundle over the unit tangent space over Ω generalizing the work of Goldman-Labourie-Margulis on complete flat Lorentz 3-manifolds. We end with showing that a T-end has a CA-lens neighborhood if it satisfies the uniform middle eigenvalue condition. In Chapter 6, we start to study the properly convex R-end theory. First, we discuss the holonomy representation spaces. Tubular actions and the dual theory of affine actions are discussed. We show that distanced actions and asymptotically nice actions are dual. We explain that the uniform middle eigenvalue condition implies the existence of the distanced action. The main result here is the characterization of R-ends whose end holonomy groups satisfy uniform middle eigenvalue conditions. That is, they are either lens-shaped R-ends. Here, we will classify R-ends satisfying the uniform middle eigenvalue conditions. We also discuss some important properties of lens-shaped R-ends. Finally, we will show that lens-shaped T-ends and lens-shaped R-ends are dual to each other. We end with discussing the properties of T-ends as obtained by this duality. In Chapter 7, we investigate the applications of the radial end theory such as the stability condition. We discuss the expansion and shrinking of the end neighborhoods. Finally, we will prove Theorem 1.21, the strong irreducibility of strongly tame properly convex orbifolds with generalized-lens shaped ends or horospherical ends. In Chapter 8, we discuss the R-ends that are NPNC. First, we show that the end holonomy group for an NPNC-end E will have an exact sequence 1 N h(π 1 (Ẽ)) N K 1 where N K is in the projective automorphism group Aut(K) of a properly convex compact set K, N is the normal subgroup of elements mapping to the trivial automorphism of K, and K o /N K is compact. We show that ΣẼ is foliated by complete affine subspaces of dimension 1. We explain that an NPNC-end satisfying the transverse weak middle eigenvalue condition for NPNC-ends is a quasi-joined R-end. We note that Chapters 5, 6, and 8, we work in the S n -versions only.

96 CHAPTER 4 Introduction to the theory of convex radial ends: classifying complete affine ends In Section 4.1, we will discuss the convex radial ends of orbifolds, covering most elementary aspects of the theory. For a properly convex real projective orbifold, the space of rays for each R-end give us a closed real projective orbifold of dimension n 1. The orbifold is convex. The universal cover can be a complete affine subspace (CA) or a properly convex domain (PC) or a convex domain that is neither (NPNC). We discuss objects associated with R-ends, and examples of ends; horospherical ones, totally geodesic ones, and bendings of ends to obtain more general examples of ends. In Section 4.2, we discuss horospherical ends. We show that a pre-horospherical ends are horospherical ends. First, horospherical ends are complete ends and have holonomy matrices with only unit norm eigenvalues and their end fundamental groups are virtually abelian. Conversely, a complete end in a properly convex orbifold has to be a horospherical end or another type that we can classify. For some proofs here we need results of Chapter R-Ends In this section, we begin by explaining the R-ends. We classify them into three cases: complete affine ends, properly convex ends, and nonproperly convex and not complete affine ends. We also introduce T-ends. Recall that an R-p-end Ẽ is convex if ΣẼ is convex. Since ΣẼ is a convex open domain and hence is contractible by Proposition 2.8, it always lifts to S n as an embedding, a convex R-end is either (i): complete affine (CA), (ii): properly convex (PC), or (iii): convex but not properly convex and not complete affine (NPNC) Examples of ends. We will present some examples here, which we will fully justify later. Recall the Klein model of hyperbolic geometry: It is a pair (B,Aut(B)) where B is the interior of an ellipsoid in RP n or S n and Aut(B) is the group of projective automorphisms of B. Now, B has a Hilbert metric which in this case is the hyperbolic metric times a constant. Then Aut(B) is the group of isometries of B. (See Section ) From hyperbolic manifolds, we obtain some examples of ends. Let M be a complete hyperbolic manifold with cusps. M is a quotient space of the interior B of an ellipsoid in RP n or S n under the action of a discrete subgroup Γ of Aut(B). Then horoballs are p-end neighborhoods of the horospherical R-ends. We generalize Definition 3.8. Suppose that a noncompact strongly tame convex real projective orbifold M has totally geodesic embedded surfaces S 1,..,S m homotopic to the ends. Let M be covered by a properly convex domain M in an affine subspace of S n. 81

97 824. INTRODUCTION TO THE THEORY OF CONVEX RADIAL ENDS: CLASSIFYING COMPLETE AFFINE ENDS We remove the outside of S j s to obtain a properly convex real projective orbifold M with totally geodesic boundary. Suppose that each S j can be considered a lens-shaped T-end. Each S i corresponds to a disjoint union of totally geodesic domains j J S i, j in M for a collection J. For each S i, j M, a group Γ i, j acts on it where S i, j /Γ i, j is a closed orbifold projectively diffeomorphic to S i. Suppose that Γ i, j fixes a point p i, j outside M. Hence, we form the cone M i, j := {p i, j } S i, j. We obtain the quotient M i, j /Γ i, j {p i, j } and identify S i, j /Γ i, j to S i, j in M to obtain the examples of real projective manifolds with R-ends. ({p i, j } S i, j ) o is an R-p-end neighborhood and the end is a totally geodesic R- end. The result is convex by Lemma 10.3 since we can think of S j as an ideal boundary component of M and that of M i, j /Γ i, j {p i, j }. This orbifold is called the hyperideal extension of the convex real projective orbifold as a convex real projective orbifold. When M is hyperbolic, each S j is lens-shaped by Proposition 4.1. Hence, the hyperideal extension of hyperbolic orbifolds are properly convex. PROPOSITION 4.1. Suppose that M is a strongly tame convex real projective orbifold. Let Ẽ be an R-p-end of M. Suppose that the p-end holonomy group of π 1 (Ẽ) is generated by the homotopy classes of finite orders or is generated by elements whose holonomy transformation fixes the p-end vertex with eigenvalues 1 or equivalently satisfies the unit middle eigenvalue condition and Ẽ has a π 1 (Ẽ)-invariant n 1-dimensional totally geodesic properly convex domain D in a p-end neighborhood and not containing the p-end vertex in the closure of D. Then the R-p-end Ẽ is lens-shaped. PROOF. Let M be the universal cover of M in S n. Ẽ is an R-p-end of M, and Ẽ has a π 1 (Ẽ)-invariant n 1-dimensional totally geodesic properly convex domain D. Since D projects to ΣẼ, D is transverse to radial lines from vẽ. Under the first assumption, since the end holonomy group ΓẼ is generated by elements of finite order, the eigenvalues of the generators corresponding to the p-end vertex vẽ equal 1 and hence every element of the end holonomy group has 1 as the eigenvalue at vẽ. Now assume that the the end holonomy groups fix the p-end vertices with eigenvalues equal to 1. Then the p-end neighborhood U can be chosen to be the open cone over the totally geodesic domain with vertex vẽ. U is projectively diffeomorphic to the interior of a properly convex cone in an affine subspace A n. The end holonomy group acts on U as a discrete linear group of determinant 1. The theory of convex cones applies, and using the level sets of the Koszul-Vinberg function, we obtain a one-sided convex neighborhood N in U with smooth boundary (see Lemmas 6.5 and 6.6 of Goldman [74]). Let F be a fundamental domain of N with a compact closure in O. We obtain a one-sided neighborhood in the other side as follows: We take R(N) for by a reflection R fixing each point of the hyperspace containing Σ and the p-end vertex. Then we choose a diagonalizable transformation D fixing the p-end vertex and every point

98 4.1. R-ENDS 83 of Σ so that the image D R(F) is in O. It follows that D R(N) O as well. Thus, N D R(N) is the CA-lens we needed. The interior of the cone vẽ (N D R(N)) is the lens-cone neighborhood for Ẽ. [SS n ] We remark that Proposition 4.1 also follows by Lemma However, we used more elementary results to prove it here. A more specific example is below. Let S 3,3,3 denote the 2-orbifold with base space homeomorphic to a 2-sphere and three cone-points of order 3. The orbifolds satisfying the following properties are the example of Tillman [123]. the hyperbolic Coxeter 3-orbifolds based on an ideal 3-polytopes of dihedral angles π/3. (See Choi-Hodgson-Lee [50].) The following is more specific version of Lemma We give a more elementary proof not depending on the full theory of this monograph. PROPOSITION 4.2. Let O be a strongly tame convex real projective 3-orbifold with R-ends where each end orbifold is diffeomorphic to a sphere S 3,3,3 or a disk with three silvered edges and three corner-reflectors of orders 3, 3, 3. Assume that the holonomy group of π 1 (O) is strongly irreducible. Then the orbifold has only lens-shaped R-ends or horospherical R-ends. PROOF. Again, it is sufficient to prove this for the case O S 3. Let Ẽ be an R-p-end corresponding to an R-end whose end orbifold is diffeomorphic to S 3,3,3. It is sufficient to consider only S 3,3,3 since it double-covers the disk orbifold. Since ΓẼ is generated by finite order elements fixing a p-end vertex vẽ, every holonomy element has eigenvalue equal to 1 at vẽ. Take a finite-index free abelian group A of rank two in ΓẼ. Since Σ E is convex, a convex projective torus T 2 covers Σ E finitely. Therefore, ΣẼ is projectively diffeomorphic either to a complete affine subspace or the interior of a properly convex triangle or a half-space by the classification of convex tori found in many places including [74] and [11] and Proposition Since there exists a holonomy automorphism of order 3 fixing a point of ΣẼ, it cannot be a quotient of a half-space with a distinguished foliation by lines. Thus, the end orbifold admits a complete affine structure or is a quotient of a properly convex triangle. Suppose that ΣẼ has a complete affine structure. Since λ vẽ (g) = 1 for all g ΓẼ, the only possibility from Theorem 4.8 is when ΓẼ is virtually nilpotent and and we have a horospherical p-end for Ẽ. Suppose that ΣẼ has a properly convex open triangle T as its universal cover. A acts with an element g with the largest eigenvalue > 1 and the smallest eigenvalue < 1 as a transformation in SL ± (3,R) the group of projective automorphisms at S 2 vẽ. As an element of SL ± (4,R), we have λ vẽ (g ) = 1 and the product of the remaining eigenvalues is 1, the corresponding the largest and smallest eigenvalues are > 1 and < 1. Thus, an element of SL ± (4,R), g fixes v 1 and v 2 other than vẽ in directions of vertices of T. Since ΓẼ has an order three elements exchanging the vertices of T, there are three fixed points of an element of A different from vẽ,vẽ. By commutativity, there is a properly convex compact triangle T S 3 with these three fixed points where A acts on. Hence, A is diagonalizable over the reals. We can make any vertex of T to be an attracting fixed point of an element of A. Each element g ΓẼ conjugates elements of A to A. Therefore g sends the attracting fixed points of elements of A to those of elements of A. Hence g(t ) = T for all g ΓẼ.

99 844. INTRODUCTION TO THE THEORY OF CONVEX RADIAL ENDS: CLASSIFYING COMPLETE AFFINE ENDS Each point of the edge E of Cl(T ) is an accumulation point of an orbit of A by taking a sequence g i so that the associated eigenvalues λ 1 (g i ) and λ 2 (g i ) are going to + while log λ 1 (g)/λ 2 (g) is bounded. Since λ vẽ = 1, writing every vector as a linear combinations of vectors in the direction of the four vectors, this follows. Hence bdt bdo and T Cl(O). If T o bdo /0, then T bdo by Lemma Then each segment from vẽ ending in bdo is in the direction of Cl(ΣẼ) = T. It must end at T. Hence, O = (T vẽ) o, an open tetrahedron σ. Since the holonomy group acts on it, we can take a finite-index group fixing each vertex of σ. Thus, the holonomy group is virtually reducible. This is a contradiction. Therefore, T O as T bdo = /0. We have a totally geodesic R-end, and by Proposition 4.1, the end is lens-shaped. (See also [31].) [SS n ] The following construction is called bending and was investigated by Johnson and Millson [90]. These give us examples of R-ends that are not totally geodesic R-ends. See Ballas and Marquis [8] for other examples. EXAMPLE 4.3 (Bending). Let O have the usual assumptions. We will concentrate on an end and not take into consideration of the rest of the orbifold. Certainly, the deformation given here may not extend to the rest. (If the totally geodesic hypersurface exists on the orbifold, the bending does extend to the rest.) Suppose that O is oriented hyperbolic manifold with a hyperideal end E. Then E is a totally geodesic R-end with an R-p-end Ẽ. Let the associated orbifold Σ E for E of O be a closed 2-orbifold and let c be a two-sided simple closed geodesic in Σ E. Suppose that E has an open end neighborhood U in O diffeomorphic to Σ E (0,1) with totally geodesic bdu diffeomorphic to Σ E. Let Ũ be a p-end neighborhood in O corresponding to Ẽ bounded by ΣẼ covering Σ E. Then U has a radial foliation whose leaves lifts to radial lines in Ũ from vẽ. Let A be an annulus in U diffeomorphic to c (0,1), foliated by leaves of the radial foliation of U. Now a lift c of c is in an embedded disk A, covering A. Let g c be the deck transformation corresponding to c and c. Suppose that g c is orientation-preserving. Since g c is a hyperbolic isometry of the Klein model, the holonomy g c is conjugate to a diagonal matrix with entries λ,λ 1,1,1, where λ > 1 and the last 1 corresponds to the vertex vẽ. We take an element k b of SL ± (4,R) of form in this system of coordinates (4.1) b 1 where b R. k b commutes with g c. Let us just work on the end E. We can bend E by k b : Now, k b induces a diffeomorphism ˆk b of an open neighborhood of A in U to another one of A since k b commutes with g c. We can find tubular neighborhoods N 1 of A in U and N 2 of A. We choose N 1 and N 2 so that they are diffeomorphic by a projective map ˆk b. Then we obtain two copies A 1 and A 2 of A by completing U A. Give orientations on A and U. Let N 1, denote the left component of N 1 A and let N 2,+ denote the right component of N 2 A. We take a disjoint union (U A) N 1 N 2 and quotient it by identifying the real projectively diffeomorphic copy of N 1, in N 1 with N 1, in U A by the identity map and identify the real projectively diffeomorphic copy of N 2,+ in N 2 with N 2,+ in U A by the identity also. We glue back N 1 and N 2 by the real projective diffeomorphism ˆk b

100 4.2. CHARACTERIZATION OF COMPLETE R-ENDS 85 of a neighborhood of N 1 to that of N 2. Then N 1 (N 1, A) is identified with N 2,+ and N 2 (N 2,+ A) is identified with N 1,. We obtain a new manifold. For sufficiently small b, we see that the end is still lens-shaped. and it is not a totally geodesic R-end. (This follows since the condition of being a lens-shaped R-end is an open condition. See Section 11.2.) For the same c, let k s be given by (4.2) s s s /s 3 where s R +. These give us bendings of the second type. For s sufficiently close to 1, the property of being lens-shaped is preserved and being a totally geodesic R-end. (However, these will be understood by cohomology.) If sλ < 1 for the maximal eigenvalue λ of a closed curve c 1 meeting c odd number of times, we have that the holonomy along c 1 has the attracting fixed point at vẽ. This implies that we no longer have lens-shaped R-ends if we have started with a lens-shaped R-end Characterization of complete R-ends The results here overlap with the results of Crampon-Marquis [61] and Cooper-Long- Tillman [60]. However, the results are of different direction than theirs and were originally conceived before their papers appeared. We also make use of Crampon-Marquis [62]. Let Ẽ be an R-p-end. A middle eigenvalue condition for a p-end holonomy group h(π 1 (Ẽ)) with respect to vẽ or the R-p-end structure holds if for each g h(π 1 (Ẽ)) {I} the largest norm λ 1 (g) of eigenvalues of g is strictly larger than the eigenvalue λ vẽ (g) associated with p-end vertex vẽ. Given an element g h(π 1 (Ẽ)), let ( λ1 (g),..., λ n+1 (g)) be the (n + 1)-tuple of the eigenvalues listed with multiplicity given by the characteristic polynomial of g where we repeat each eigenvalue with the multiplicity given by the characteristic polynomial. The multiplicity of a norm of an eigenvalues of g is the number of times the norm occurs among the (n + 1)-tuples of norms ( ) λ 1 (g),..., λ n+1 (g). DEFINITION 4.4. A weak middle eigenvalue condition for an R-p-end holonomy group h(π 1 (Ẽ)) holds if for each g h(π 1 (Ẽ)) the eigenvalue λ vẽ associated with the p- end vertex vẽ(g) or the R-p-end structure has the largest norm of all norms of eigenvalues of elements of h(π 1 (Ẽ)), then the norm of the eigenvalue must have multiplicity 2. We note that these definitions depend on the choice of the p-end vertices; however, they are well defined once the radial structures are assigned. Recall the parabolic subgroup of the isometry group Aut(B) of the hyperbolic space B for an (i 0 + 1)-dimensional Klein model B S i 0+1 fixing a point p in the boundary of B. Such a discrete subgroup of a parabolic subgroup group is isomorphic to extensions of a lattice in R i 0 and is Zariski closed by the Bieberbach theorem. Let E be an i 0 -dimensional ellipsoid containing the point v in a subspace P of dimension i in S n. Let Aut(P) denote the group of projective automorphisms of P, and let SL ± (n + 1,R) P the subgroup of SL ± (n + 1,R) acting on P. Let r P : SL ± (n + 1,R) P Aut(P) denote the restriction homomorphism g g P. An i 0 -dimensional partial cusp

101 864. INTRODUCTION TO THE THEORY OF CONVEX RADIAL ENDS: CLASSIFYING COMPLETE AFFINE ENDS subgroup is one mapping under R P isomorphically to a cusp subgroup of Aut(P) acting on E {v}, fixing v. Suppose now that O RP n. Let P denote a subspace of dimension i containing an i 0 -dimensional ellipsoid E containing v. Let PGL(n + 1,R) P denote the subgroup of PGL(n+1,R) acting on P. Let R P : PGL(n+1,R) P Aut(P ) denote the restriction g g P. An i 0 -dimensional partial cusp subgroup is one mapping under R P isomorphically to a cusp subgroup of Aut(P ) acting on E {v}, fixing v. When i 0 = n 1, we will drop the partial from the term partial cusp group. An i 0 -dimensional cusp group is a finite extension of a projective conjugate of a discrete cocompact subgroup of a group of an i 0 -dimensional partial cusp subgroup. If the horospherical neighborhood with the p-end vertex v has the p-end holonomy group that is a discrete (n 1)-dimensional cusp group, then we call the p-end to be cusp-shaped or horospherical. (See Theorem 4.7.) Our first main result classifies CA R-p-ends. We need the notion of NPNC-ends that will be explained in Section 8.2. Given a hospherical R-p-end, the p-end holonomy group Γ v acts on a p-end neighborhood U and Γ v is a subgroup of an (n 1)-dimensional cusp group H v. By Lemma 2.15, V := g(u) g H v contains a H v -invariant p-end neighborhood. Hence, V is a horospherical p-end neighborhood of Ẽ. Thus, a horospherical R-end is pre-horospherical. Conversely, a prehorospherical R-end is a horospherical R-end by Theorem 4.7 under some assumption on O itself. (See Definition 1.7.) COROLLARY 4.5. Let O be a strongly tame properly convex real projective n-orbifold. Let E be an R-end of its universal cover O. Then E is a pre-horospherical R-end if and only if Ẽ is a horospherical R-end. THEOREM 4.6. Let O be a strongly tame properly convex real projective n-orbifold. Let Ẽ be an R-p-end of its universal cover O. Let ΓẼ denote the end holonomy group. Then Ẽ is a complete affine R-p-end if and only if Ẽ is a horospherical R-p-end or an NPNC-end with fibers of dimension n 2 by altering the p-end vertex. Furthermore, if Ẽ is a complete affine R-p-end and ΓẼ satisfies the weak middle eigenvalue condition, then Ẽ is a horospherical R-p-end. PROOF. Again, we assume that Ω is a domain of S n. Theorem 4.8 is the forward direction. Corollary 8.41 implies the second case above. Now, we prove the converse using the notation and results of Chapter 8. (The reader may need to defer reading the proof to after reading Chapter 8.) Since a horospherical R-p-end is pre-horospherical, Theorem 4.7 implies the half of the converse. Given a p- NPNC-end Ẽ with fibers of dimension n 2, ΣẼ is projectively isomorphic to an affine half-space. Using the notation of Proposition 8.27, K is zero-dimensional and the end holonomy group ΓẼ acts on K v for an end vertex v. The space of leaves is projectively diffeomorphic to the interior of K v. Now we switch the p-end vertex to a singleton {k } = K. Then each leaf projects to an ellipsoid in S n 1 k with a common vertex v corresponding to k v oriented away from k. The ellipsoids are tangent to a common hyperspace in S n 1 k. Hence, they are parallel paraboloids in an affine subspace A n 1 S n 1 k. The uniform positive translation condition gives us that the union of the parallel paraboloids is

102 4.2. CHARACTERIZATION OF COMPLETE R-ENDS 87 A n 1. Hence, Ẽ is a complete R-end with k as the vertex. The last statement follows by Corollary [SS n ] THEOREM 4.7 (Horosphere). Let O be a strongly tame convex real projective n- orbifold. Let Ẽ be a pre-horospherical R-end of its universal cover O, O S n (resp. RP n ) and ΓẼ denote the p-end holonomy group. Then the following hold: (i) The space ΣẼ = R vẽ ( O) S n 1 vẽ of equivalence classes of lines segments from the endpoint vẽ in O forms a complete affine subspace of dimension n 1. (ii) The norms of eigenvalues of g ΓẼ are all 1. (iii) ΓẼ virtually is in a conjugate of an abelian parabolic or cusp subgroup of SO(n,1) (resp. PO(n,1)) of rank n 1 in SL ± (n + 1,R) (resp. PGL(n + 1,R)). And hence Ẽ is cusp-shaped. (iv) For any compact set K O, O contains a horospherical end neighborhood disjoint from K. (v) A p-end vertex of a horospherical p-end cannot be an endpoint of a segment in bdo. PROOF. We will prove for the case O S n. Let U be a horospherical p-end neighborhood with the p-end vertex vẽ. The space of great segments from the p-end vertex passing U forms a convex subset ΣẼ of a complete affine subspace R n 1 S n 1 by Proposition The space covers an end orbifold ΣẼ with the discrete group π 1 (Ẽ) acting Ẽ as a discrete subgroup Γ Ẽ of the projective automorphisms so that ΣẼ/Γ is projectively Ẽ isomorphic to ΣẼ. (i) By Proposition 2.39, one of the following three happens: ΣẼ is properly convex. ΣẼ is foliated by complete affine subspaces of dimension i 0, 1 i 0 < n 1, with the common boundary sphere of dimension i 0 1 and the space of the leaves forms a properly open convex subset K o of S n i 0 1. ΓẼ acts on K o cocompactly but perhaps not discretely. ΣẼ is a complete affine subspace. We aim to show that the first two cases do not occur. Suppose that we are in the second case and 1 i 0 n 2. This implies that ΣẼ is foliated by complete affine subspaces of dimension i 0 n 2. Since ΓẼ acts on a properly convex subset K of dimension 1, an element g has a norm of an eigenvalue > 1 and a norm of eigenvalue < 1 by Proposition 1.1 of [13] as a projective automorphism on the great sphere spanned by K. Hence, we obtain the largest norm of eigenvalues and the smallest one of g in Aut(S n ) both different from 1. By Lemma 2.27, g is bi-semi-proximal. Therefore, let λ 1 (g) > 1 be the greatest norm of the eigenvalues of g and λ 2 (g) < 1 be the smallest norm of the eigenvalues of g as an element of SL ± (n + 1,R). Let λ vẽ (g) > 0 be the eigenvalue of g associated with vẽ. These are all positive. The possibilities for g are as follows λ 1 (g)= λ vẽ (g) > λ 2 (g), λ 1 (g)> λ vẽ (g) > λ 2 (g), λ 1 (g) > λ 2 (g)= λ vẽ (g).

103 884. INTRODUCTION TO THE THEORY OF CONVEX RADIAL ENDS: CLASSIFYING COMPLETE AFFINE ENDS In all cases, at least one of the largest norm or the smallest norm is different from λ vẽ (g). By Lemma 2.27, this norm is realized by a positive eigenvalue. We take g n (x) for a generic point x U. As n or n, the sequence limits to a point x in Cl(U) distinct from vẽ. Also, g fixes a point x, and x has a different positive eigenvalue from λ vẽ (g). As x U, it should be x = vẽ by the definition of the pre-horospheres. This is a contradiction. The first possibility is also shown not to occur similarly. Thus, ΣẼ is a complete affine subspace. (ii) If g ΓẼ has a norm of eigenvalue different from 1, then we can apply the second and the third paragraphs above to obtain a contradiction. We obtain λ j (g) = 1 for each norm λ j (g) of eigenvalues of g for every g ΓẼ. (iii) Since ΣẼ is a complete affine subspace, ΣẼ/ΓẼ is a complete affine orbifold with the norms of eigenvalues of holonomy matrices all equal to 1 where Γ Ẽ denotes the affine transformation group corresponding to ΓẼ. (By D. Fried [68], this implies that π 1 (Ẽ) is virtually nilpotent.) Again by Selberg Theorem 2.11, we can find a torsion-free subgroup Γ Ẽ of finite-index. Then Γ Ẽ is in a cusp group by Proposition 7.21 of [61] (related to Theorem 1.6 of [61]). By the proposition, we see that Γ Ẽ is in a conjugate of a parabolic subgroup of SO(n,1) and hence acts on an (n 1)-dimensional ellipsoid fixing a unique point. Since a horosphere has a Euclidean metric invariant under the group action, the image group is in a Euclidean isometry group. Hence, the group is virtually abelian by the Bieberbach theorem. Actually, there is a one-dimensional family of such ellipsoids containing the fixed point where Γ Ẽ acts on. Let U denote the domain bounded by the closure of the ellipsoid. There exist finite elements g 1,...,g n representing cosets of ΓẼ/Γ Ẽ. If g i(u) is a proper subset of U, the g n i (U) is so and hence gn i is not in Γ Ẽ for any n. This is a contradiction. Hence ΓẼ acts on U also. By same reasoning, ΓẼ on every ellipsoid in a one-dimensional parameter space containing a unique fixed point, and an ellipsoid gives us a horosphere in the interior of a horoball. Hence, ΓẼ is a cusp group. (iv) We can choose an exiting sequence of p-end horoball neighborhoods U i where a cusp group acts. We can consider the hyperbolic spaces to understand this. (v) Suppose that bdo contains a segment s ending at the p-end vertex vẽ. Then s is on an invariant hyperspace of ΓẼ. Now conjugating ΓẼ into an (n 1)-dimensional parabolic or cusp subgroup P of SO(n,1) fixing (1, 1,0,...,0) R n+1 by say an element h of SL ± (n + 1,R). By simple computations using the matrix forms of ΓẼ, we can find a sequence g i hγẽh 1 P so that {g i (h(s))} geometrically converges to a great segment. Thus, for a sequence h 1 g i h ΓẼ, h 1 g i h(s) geometrically converges to a great segment in Cl( O). This contradicts the proper convexity of O. [S n T] We will now show the converse of Theorem 4.7. THEOREM 4.8 (Complete affine). Let O be a strongly tame properly convex n-orbifold. Suppose that Ẽ is a complete-affine R-p-end of its universal cover O in S n ( resp. in RP n ). Let vẽ S n ( resp. RP n ) be the p-end vertex with the p-end holonomy group ΓẼ. Then (i) ΓẼ is virtually unipotent where all norms of eigenvalues of elements equal 1, or ΓẼ is virtually nilpotent where each g ΓẼ has at most two norms of the eigenvalues, (A finite-index subgroup has only two eigenvalues which are positive) at least one g ΓẼ has two norms, and

104 4.2. CHARACTERIZATION OF COMPLETE R-ENDS 89 if g ΓẼ has two distinct norms of the eigenvalues, the norm of λ vẽ (g) has a multiplicity one. (ii) In the first case, ΓẼ is horospherical, i.e., cuspidal. PROOF. We first prove for the S n -version. Using Theorem 2.11, we may choose a torsion-free finite-index subgroup. We may assume without loss of generality that Γ is torsion-free since we only need to prove the theorem for a finite-index subgroup. Hence, Γ does not fix a point in ΣẼ. (i) Since Ẽ is complete affine, ΣẼ S n 1 vẽ is identifiable with an affine subspace R n 1. ΓẼ induces Γ Ẽ in Aff(Rn 1 ) that are of form x Mx+ b where M is a linear map R n 1 R n 1 and b is a vector in R n 1. For each γ ΓẼ, let γ R n 1 denote this affine transformation, we denote by ˆL(γ R n 1) the linear part of the affine transformation γ R n 1, and let v(γ R n 1) denote the translation vector. A relative eigenvalue is an eigenvalue of ˆL(γ R n 1). At least one eigenvalue of ˆL(γ R n 1) is 1 since γ acts without fixed point on R n 1. (See [98].) Now, ˆL(γ R n 1) has a maximal invariant vector subspace A of R n 1 where all norms of the eigenvalues are 1. Suppose that A is a proper γ-invariant vector subspace of R n 1. Then γ R n 1 acts on the affine space R n 1 /A as an affine transformation with the linear parts without a norm of eigenvalue equal to 1. Hence, γ R n 1 has a fixed point in R n 1 /A, and γ R n 1 acts on an affine subspace A parallel to A. A subspace H containing vẽ corresponds to the direction of A from vẽ. The union of segments with endpoints vẽ,vẽ in the directions in A S n 1 vẽ is in an open hemisphere of dimension < n. Let H + denote this space where bdh + vẽ holds. Since ΓẼ acts on A, it follows that ΓẼ acts on H +. Then γ has at most two eigenvalues associated with H + one of which is λ v (γ) and the other is to be denoted λ + (γ). Since γ fixes vẽ and there is an eigenvector in the span of H + associated with λ + (γ), γ has the matrix form where we have γ = λ + (γ)ˆl(γ R n 1) λ + (γ) v(γ R n 1) 0 0 λ + (γ) 0 λ vẽ (γ) λ + (γ) n det(ˆl(γ R n 1))λ vẽ (γ) = ±1. (Note λ vẽ (γ 1 ) = λ vẽ (γ) 1 and λ + (γ 1 ) = λ + (γ) 1.) We will show that ˆL(γ R n 1) for every γ ΓẼ is unit-norm-eigenvalued below. There are following possibilities for each γ ΓẼ: (a) λ 1 (γ) > λ + (γ) and λ 1 (γ) > λ vẽ (γ). (b) λ 1 (γ) = λ + (γ) = λ vẽ (γ). (c) λ 1 (γ) = λ + (γ),λ 1 (γ) > λ vẽ (γ). (d) λ 1 (γ) > λ + (γ),λ 1 (γ) = λ vẽ (γ). Suppose that γ satisfies (b). The relative eigenvalues of γ on R n 1 are all 1. Either γ is unit-norm-eigenvalued or we can take γ 1 and we are in case (a). Suppose that γ satisfies (a). There exists a projective subspace S of dimension 0 where the points are associated with eigenvalues with the norm λ 1 (γ) where λ 1 (γ) > λ + (γ),λ vẽ (γ).

105 904. INTRODUCTION TO THE THEORY OF CONVEX RADIAL ENDS: CLASSIFYING COMPLETE AFFINE ENDS Let S be the smallest subspace containing H and S. Let U be a p-end neighborhood of Ẽ. Since the space of directions of U is R n 1, we have U S /0. We can choose two generic points y 1 and y 2 of U S H so that y 1 y 2 meets H in its interior. Then we can choose a subsequence m i, m i, so that γ m i(y 1 ) f and γ m i(y 2 ) f as i + unto relabeling y 1 and y 2 for a pair of antipodal points f, f S. This implies f, f Cl( O), and O is not properly convex, which is a contradiction. Hence, (a) cannot be true. We showed that if any γ ΓẼ satisfies (a) or (b), then γ is unit-norm-eigenvalued. If γ satisfies (c), then (4.3) λ 1 (γ) = λ + (γ) λ i (γ) λ vẽ (γ) for all other norms of eigenvalues λ i (γ): Otherwise, γ 1 satisfies (a), which cannot happen. Similarly if γ satisfies (d), then we have (4.4) λ 1 (γ) = λ vẽ (γ) λ i (γ) λ + (γ) for all other norms of eigenvalues λ i (γ). We conclude that either γ is unit-norm-eigenvalued or satisfies (4.3) or (4.4). There is a homomorphism This gives us an exact sequence λ vẽ : ΓẼ R + given by g λ vẽ (g). (4.5) 1 N ΓẼ R 1 where R is a finitely generated subgroup of R +, an abelian group. For an element g N, λ vẽ (g) = 1. Since the relative eigenvalue corresponding to ˆL(g R n 1) A is 1, a matrix form shows that λ + (g) = 1 for g N. (4.3) and (4.4) and the conclusion of the above paragraph show that g is unit-norm-eigenvalued. Thus, N is therefore virtually nilpotent by Theorem (See Fried [68]). Taking a finite cover again, we may assume that N is nilpotent. Since R is a finitely generated abelian group, ΓẼ is solvable by (4.5). Since ΣẼ = R n 1 is complete affine, Proposition S of Goldman and Hirsch [77] implies det(g R n 1) = 1 for all g ΓẼ. If γ satisfies (c), then all norms of eigenvalues of γ except for λ vẽ (γ) equal λ + (γ) since otherwise by (4.3), relative eigenvalues satisfy λ i (γ)/λ + (γ) 1, and the above determinant is less than 1. Similarly, if γ satisfies (d), then similarly all norms of eigenvalues of γ except for λ vẽ (γ) equals λ + (γ). Therefore, only (b), (c), and (d) hold and g R n 1 is unit-norm-eigenvalued for all g ΓẼ. By Theorem 2.25, ΓẼ R n 1 is an orthopotent group and hence is virtually unipotent by Theorem 3 of Fried [68]. Now we go back to ΓẼ. Suppose that every γ is orthopotent. Then we have the first case of (i). If not, then the second case of (i) holds. (ii) This follows by Lemma 4.9. [S n T] The second case will be studied later in Corollary We will show the end Ẽ to be a NPNC-end with fiber dimension n 2 when we choose another point as the new p-end vertex for Ẽ. Clearly, this case is not horospherical. (See Crampon-Marquis [61] for a similar proof.)

106 4.2. CHARACTERIZATION OF COMPLETE R-ENDS 91 LEMMA 4.9. Assume that O is a properly convex real projective orbifolds with an end E. Suppose that E is a convex end with a corresponding p-end Ẽ. Suppose that eigenvalues of elements of ΓẼ have unit norms only. Then ΓẼ is conjugate to a subgroup of a parabolic subgroup in SO(n,1) or PO(n,1) and a finite-index subgroup of ΓẼ is nilpotent and Ẽ is horospherical, i.e., cuspidal. PROOF. We will assume first O S n. By Theorem 2.25, ΓẼ is virtually orthopotent. By Proposition 4.11, ΣẼ is complete affine, and ΓẼ acts on it as an affine transformation group. By Theorem 3 in Fried [68], ΓẼ is virtually unipotent. Since ΣẼ/ΓẼ is a compact complete-affine manifold, a finite-index subgroup F of ΓẼ is contained in a unipotent Lie subgroup acting on ΣẼ. Now, by Malcev [112], it follows that the same group is contained in a simply connected unipotent group N acting on S n since F is unipotent. The dimension of N is n 1 = dim ΣẼ by Theorem 3 of [68]. Let U be a component of the inverse image of a p-end neighborhood so that vẽ bdu. Assume that U is a radial p-end neighborhood of vẽ. The group N acts on a smaller open set covering a p-end neighborhood by Lemma We let U be this open set from now on. Consequently, bdu O is smooth. Now N acts properly on ΣẼ by Lemma 4.10 since F acts properly on it and N/F is compact. N acts cocompactly on ΣẼ since so does F, F N. By Proposition 4.13, N is a cusp group, and U is a p-end neighborhood bounded by an ellipsoid. Since ΓẼ has a finite extension of N as the Zariski closure, the connected identity component N is normalized by ΓẼ. Also, for element g ΓẼ F, suppose g(x) U. Now, g(bdu {vẽ}) is an orbit of g(x) for x bdu {vẽ}). Hence, g(bdu {vẽ})) U. Hence g n is not in F for all n, a contradiction. Also, g(x) cannot be outside Cl(U) similarly. Hence, ΓẼ acts on U. Also, ΓẼ is in a conjugate of a parabolic subgroup. [S n T] LEMMA Suppose that a closed connected projective group G acts properly and cocompactly on a convex domain Ω in S n (resp. RP n ). Then G acts transitively on Ω. (Lemma 2.5 of [16]). Suppose that Γ is a uniform lattice in a closed connected group G acting on a submanifold Ω in S n ( resp. RP n ). Suppose that Γ acts properly and cocompactly on Ω. Then G acts transitively on Ω. PROOF. For the second item, we claim that G acts properly also. Let F be the fundamental domain of G with Γ action. Let x Ω. Let F be the image F(x) := {g(x) g F} in Ω. This is a compact set. Define Γ F := {g Γ g(f(x)) F(x) /0}. Then Γ F is finite by the properness of the action of Γ. Since an element of G is a product of an element g of Γ and f F, and g f (x) = x, it follows that g F(x) F(x) /0 and g Γ F. Hence the stabilizer G x Γ F F, and G x is compact. The second part follows from the first part since G must act properly and cocompactly. [S n T] PROPOSITION Let N be a discrete group or an n 1-dimensional connected Lie group where all the elements have only eigenvalues of unit norms acting on a convex n 1-domain Ω projectively and properly and cocompactly. Then Ω is a complete affine space. If N is a connected Lie group, then N is a simply-connected unit-norm-eigenvalued solvable group.

107 924. INTRODUCTION TO THE THEORY OF CONVEX RADIAL ENDS: CLASSIFYING COMPLETE AFFINE ENDS PROOF. Again, we first prove for the S n -version. By Theorem 2.25, N is an orthopotent Lie group. First, Ω cannot be properly convex: By Fait 1.5 of [16], N either acts irreducibly on Ω or Ω is a join of domain Ω 1,...,Ω n where N acts irreducibly on each Ω i. Since a solvable group never acts irreducibly unless the domain is 0-dimensional by the Lie- Kolchin theorem, Ω is a simplex or a point. (See Theorem 17.6 of [88].) Then N has to be diagonalizable and this is a contradiction to the unit-norm-eigenvalued property since N acts cocompactly unless n 1 = 0. If n 1 = 0, the conclusion is true. Now suppose that Ω is not properly convex but not complete affine. Then Ω is foliated by i 0 -dimensional complete affine spaces for i 0 < n. The space of affine leaves is a properly convex domain K by discussions on R-ends in Section 4.1. Hence, N acts on K. The stabilizer N l is i 0 -dimensional since the N-action is simply transitive. Hence, N/N l acts on a properly convex set K o satisfying the premises. Again, this is a contradiction. Hence, Ω is complete affine. Now, consider the case when N is a connected Lie group. By Lemma 4.10, N acts transitively on Ω. N has an N-invariant metric on Ω by the properness of the action. Consider an orbit map N N(x) for x Ω. If a stabilizer of a point x of Ω contains a group of dimension 1, then dimn > dimω. The stabilizer is a finite group. Hence, N covers Ω finitely. Since Ω is contractible, the orbit map is a diffeomorphism. Hence, N is contractible. By Theorem 2.25, N is a solvable Lie group. [SS n ] LEMMA Let N be a closed orthopotent Lie group in SL ± (n,r) acting on R n inducing a proper action on an n 1-dimensional affine space A that is the upper halfspace of R n quotient by the scalar multiplications. Suppose that N acts cocompactly. Then there is a connected group N u acting transitively on A n 1. Moreover, N u is the unipotent subgroup in SL ± (n,r) of dimension n 1 of N normalized by N where N u /N u N is compact. PROOF. By Theorem 2.25, N is an extension of a solvable group by a compact group. We may assume without loss of generality that N is the normal solvable group. We use an induction on dimension n. For n = 1, the above is clear with N being the trivial group. Since a solvable group N has a complex basis where the group takes a triangular form by Theorem in [133], the unipotent subgroup N of N with only unit eigenvalues form a normal subgroup and N [N,N]. Thus, N/N is a unit-norm-eigenvalued connected abelian group where each nonidentity element comes from an element with at least one eigenvalue not equal to 1. Since N is unipotent, there is a unipotent Lie group N u where N u /N is compact by Malcev [112]. Since N is orthopotent, there is a flag {0} = V 0 V 1 V m = R n preserved by N where N acts as an orthogonal group on V i+1 /V i for each i = 1,...,m 1. Here, A is parallel to subspace V m 1 of dimension n 1. (See Chapter 2 of Berger [21].) This induces a parallel set of vector subspace W i on the tangent space of A with dimv i = dimw i + 1 for each i. N acts on the complete affine space A and acts on A := A/V m 1 as a Euclidean isometry group. N acts as a purely translation group since it is a unipotent Euclidean isometry group. Let W be the vector subspace generated by translation directions of N. Hence, there is a set of parallel affine subspaces parallel to the directions of N isomorphic to A := A /W where N/N acts as an abelian isometry group cocompactly.

108 4.2. CHARACTERIZATION OF COMPLETE R-ENDS 93 Suppose that A is not zero-dimensional. An abelian connected group N/N has a uniform lattice G acting purely as a cocompact translation group on an affine subspace A in A by Theorem of Bieberbach-type [124]. Since each nonidentity element of G has at least eigenvalue not equal to 1, it follows that A is a proper subspace. Moreover, A /G cannot be compact by the existence of orthogonal complement to A. This is a contradiction. Hence, A is zero-dimensional, and N acts cocompactly, and N u acts transitively. Consider affine subspaces in A parallel to V m 1. Choose one say l. There are the subgroup N l of N acting on l and the subgroup N l of N acting on it. These are solvable and unipotent respectively. Then the above discussion applies to the affine subspace l of dimension than that of A, it follows that N acts cocompactly on l/v m 2, and N u acts transitively on k/v m 2. By induction on the dimension of l, it follows that N acts cocompactly on l as well. Hence, N acts cocompactly on A, and N u acts transitively on A. We prove a stronger proposition for later purposes. PROPOSITION Let U is in an open domain in S n (resp. RP n ) radially foliated from a point p bdu with smooth bdu {p}. Suppose U is in a properly convex domain. and let N be an n 1-dimensional connected Lie group with only unit norm eigenvalues acting on U and fixing p. Suppose that it acts on R p (U) properly and cocompactly. Then U is the interior of an ellipsoid and N is a unipotent cusp group. PROOF. We first assume U S n. By Proposition 4.11, R p (U) = A is complete affine. N acts on R p (U) as a unipotent Lie group. Thus, N is a simply-connected unit-normeigenvalued solvable Lie group by Proposition We claim that N is unipotent: Since a solvable group has a complex basis where the group takes a triangular form by Theorem in [133], the subgroup N of N with only unit eigenvalues form a normal subgroup and N [N,N]. Thus, N/N is a unitnorm-eigenvalued connected abelian group where each nonidentity element comes from an element with at least one eigenvalue not equal to 1. N acts transitively and is (n 1)- dimensional by Lemma Since N N is an open set and N is connected, it follows that N = N. We will now show that U is the interior of an ellipsoid. We identify p with [1,0,...,0]. Let W denote the hyperspace in S n containing p sharply supporting U. W corresponds to of the set of directions of an open hemisphere R p (U) and hence is unique and, thus, N-invariant. Also, W Cl(U) is a properly convex subset of W. Let y be a point of U. Suppose that N contains a sequence {g i } so that a supporting hyperspace in S n 1 p g i (y) x 0 W Cl( O) and x 0 p; that is, x 0 in the boundary direction of A from p. Let U 1 = Cl(U) W. Let V be the smallest subspace containing p and U 1. The dimension of V is 1 as it contains x 0 and p. Again N acts on V. Now, V is divided into disjoint open hemispheres of various dimensions where N acts on: By Theorem of [133], N preserves a flag structure V 0 V 1 V k =V. We take components of complement V i V i 1. Let H V :=V V k 1. Suppose that dimv = n 1 for contradiction. Then H V U 1 is not empty since otherwise, we would have a smaller dimensional V. Let h V be the component of H V meeting U 1. Since N is unipotent, h V has an N-invariant metric by Theorem 3 of Fried [68].

109 944. INTRODUCTION TO THE THEORY OF CONVEX RADIAL ENDS: CLASSIFYING COMPLETE AFFINE ENDS We claim that the orbit of the action of N is of dimension n 1 and hence locally transitive on H V : If not, then a one-parameter subgroup N fixes a point of h V. This group acts trivially on h V since the unipotent group contains a trivial orthogonal subgroup. Since N is not trivial, it acts as a group of nontrivial translations on the affine subspace H o. We obtain that N (U) is not properly convex, and an orbit of N is open in h V. Hence, N acts locally simply-transitively without fixed points. The orbit of N in h V is closed since h V has an N-invariant metric. Thus, N acts transitively on h V. Hence, the orbit N(y) of N for y H V U 1 contains a component of H V. This contradicts the assumption that Cl(U) is properly convex (compare with arguments in [61].) Suppose that the dimension of V is n 2. Let J be a subspace of dimension 1 bigger than dimv and containing V and meeting U. Let J A denote the subspace of A corresponding to the directions in J. Then J is sent to disjoint subspaces or to itself under N. Since N acts on A transitively, a nilpotent subgroup N J of N acts on J A transitively. Hence, dimn J = dimj A = dimv, and we are in a situation immediately above. The orbit N J (y) for a limit point y H V contains a component of V V k 1 as above. Thus, N J (y) contains the same component, an affine subspace. As above, we have a contradiction to the proper convexity. Therefore, points such as x 0 W bd( O) {p} do not exist. Hence for any sequence of elements g i ΓẼ, we have g i (y) p. Hence, bdu = (bdu O) {p}. Clearly, bdu is homeomorphic to an (n 1)-sphere. Since U is radial, this means that U is a pre-horospherical p-end neighborhood. (See Definition 1.7.) Since N acts transitively on a complete affine space R p (U), and there is a 1 to 1 radial correspondence of R p (U) and bdu {p}, it acts so on bdu p. Since N is unipotent and acts transitively on bdu {p}, Lemma 7.12 of [61] shows that U is bounded by an ellipsoid. Choose x U, then N(x) U is an horospherical p-end neighborhood also. Since Aut(U) is the group of hyperbolic isometry group of U with the Hilbert metric, it follows that N is the cusp group. [S n T]

110 CHAPTER 5 The affine action on properly convex domain with boundary in a totally geodesic hyperspace In this chapter we will show the asymptotic niceness of the affine actions. The main tools will be Anosov flows on the affine bundle over the unit tangent bundles as in Goldman- Labourie-Margulis [78]. We will introduce a flat bundle and decompose it in an Anosovtype manner. Then we will find an invariant section. We will prove the asymptotic niceness using the sections. In Section 5.1, we wish to define affine actions and define the uniform middle eigenvalue condition. We begin first with a strictly convex domain Ω in the boundary of the affine subspace and a cocompact action. In Section 5.2, we define proximal flows and decompose the vector bundle flow into contracting and repelling and neutral subbundles. In Section 5.3, we will generalize these results to the case when Ω is not necessarily strictly convex. A basic technique here is to make the unit tangent bundle larger by blowing up using the sets of hyperspaces at the end points of geodesics. In Section 5.4, we discuss the lens condition for T-ends obtained by the uniform middle eigenvalue condition. Here we basically repeat the material for strictly convex cases since we can see the generalization in a more detailed way. Except for Section 5.4, we will work only in S n for simplicity Affine actions Let Γ be an affine group acting on the affine subspace A n with boundary bda n = S n 1 in S n, A n is an open n-hemisphere. Let U be a properly convex invariant Γ-invariant domain with the property: Cl(U ) bda n = Cl(Ω) bda n for a properly convex open domain Ω. To begin with, we assume only that Ω is properly convex. Γ is called properly convex affine action. A sharply supporting hyperspace P at x bd Cl(Ω) is asymptotic to U if there are no other sharply supporting hyperplane P at x so that P A n separates U and P A n. In this case, we say that hyperspace P is asymptotic to U. We will use the abbreviation AS-hypersurface to indicate for asymptotic sharply supporting hyperspace. A properly convex affine action of Γ is said to be asymptotically nice if Γ acts on a properly convex open domain U in A n with boundary in Ω S n 1, and Γ acts on a compact subset J := {H H is an AS- hyperspace in S n at x bdω,h S n 1 } where we require that every sharply supporting (n 2)-dimensional space of Ω in S n 1 is contained in at least one of the element of J. As a consequence, any sharply supporting (n 2)-dimensional space Q of Ω, the set H Q := {H J H Q} 95

111 96 5. PROPERLY CONVEX AFFINE ACTIONS is compact and bounded away from bda n in the Hausdorff metric d H. DEFINITION 5.1. A subspace U of R n is expanding under a linear map L if L(u) C u, C > 1, for a norm of R n. A subspace U of R n is contracting under a linear map L if L(u) C u, 0 < C < 1. for a norm of R n. This is equivalent to all the norms of eigenvalues of L U is strictly less than 1. The expanding condition is equivalent to all the norms of eigenvalues of L U is strictly larger than 1 by Corollary of Katok and Hasselblatt [95]. In this section, we will work with S n only, while the RP n versions of the results follows from the results here in an obvious manner. For each element of g Γ, (5.1) h(g) = ( 1 λẽ (g) 1/n ĥ(g) b 0 g λẽ(g) where b g is n 1-vector and ĥ(g) is an n n-matrix of determinant ±1 and λẽ(g) > 0. In the affine coordinates, it is of the form 1 (5.2) x λẽ(g) 1+ 1 n ) ĥ(g)x + 1 λẽ(g) b g. Let λ 1 (g) denote the maximal norm of the eigenvalue of g, g Γ. If there exists a uniform constant C > 1 so that (5.3) C 1 length Ω (g) log λ 1(g) λẽ(g) Clength Ω (g), g ΓẼ {I}, then Γ is said to satisfy the uniform middle eigenvalue condition with respect to the boundary hyperspace This implies that (5.4) λ 1 (g)/λẽ(g) > 1 and λ n (g)/λẽ(g) < 1. We denote by L : Aff(A n ) GL(n,R) the homomorphism τ M taking the linear part of an affine transformation τ : x Mx + b to M GL(n,R). LEMMA 5.2. Let Γ be an affine group acting on the affine subspace A n with boundary bda n in S n satisfying the uniform middle eigenvalue condition with respect to bda n. 1 Then the linear part of g equal to ĥ(g) has a nonzero expanding subspace and a contracting subspace in R n. λẽ (g) 1+ 1 n We will start with the case when Γ is hyperbolic and hence when Ω must be strictly convex with Ω being C 1 by Theorem 1.1 of [17]. THEOREM 5.3. We assume that Γ is a hyperbolic group acting on an open properly convex domain Ω in bda n with Ω/Γ is a closed real projective (n 1)-orbifold. Let Γ have a properly convex affine action on the affine subspace A n, A n S n, acting on a properly convex domain U A n so that Cl(U) bda n = Cl(Ω). Suppose that Ω/Γ is a closed (n 1)-dimensional orbifold and Γ satisfies the uniform middle-eigenvalue condition with respect to bda n. Then Γ is asymptotically nice with the properly convex open domain U, and any open set U satisfying the properties of U has the AS-hyperspace at each point of bdω is the same as that of U.

112 5.2. THE PROXIMAL FLOW. 97 In the case when the linear part of the affine maps are unimodular, Theorem of Labourie [105] shows that such a domain U exists but without showing the asymptotic niceness. The hyperbolicity of Γ shows that Ω is strictly convex by Benoist [17]. We will generalize the theorem to Theorem 5.14 without the hyperbolicity condition of Γ. Furthermore, we will show that the middle eigenvalue condition actually implies the existence of the properly convex domain U in Theorem The reason for presenting weaker Theorem 5.3 is to convey the basic idea of the proof of the generalized theorem The proximal flow The decomposition of the flow. We generalize the work of Goldman-Labourie- Margulis [78] using Anosov flows: Assume as in the premise of Theorem 5.3. Since Ω is properly convex, Ω has a Hilbert metric. Let T Ω denote the tangent space of Ω. Let UΩ denote the unit tangent bundle over Ω. This has a smooth structure as a quotient space of T Ω O/ where O is the image of the zero-section, and v w if v and w are over the same point of Ω and v = s w for a real number s > 0. We are assuming that Γ is hyperbolic. Since Σ := Ω/Γ is a strictly convex real projective orbifold, UΣ := UΩ/Γ is a compact smooth orbifold again. A geodesic flow on UΩ/Γ is Anosov and hence topologically mixing. Hence, the flow is nonwandering everywhere. (See [15].) Γ acts irreducibly on Ω, and bdω is C 1. Denote by Π Ω : UΩ Ω the projection to the base points. Let A n denote the n-dimensional affine subspace. Let h : Γ Aff(A n ) denote the representation as described in (5.2). We form the product UΩ A n that is an affine bundle over UΩ. We take the quotient à := UΩ A n by the diagonal action g(x, u) = (g(x),h(g) u) for g Γ,x UΩ, u A n. We denote the quotient by A which fibers over the smooth orbifold UΩ/Γ with fiber A n. Let V n be the vector space associated with A n. Then we can form Ṽ := UΩ V n and take the quotient under the diagonal action: g(x, u) = (g(x),l (h(g)) u) for g Γ,x UΩ, u V n where L is the homomorphism taking the linear part of g. We denote by V the fiber bundle over UΩ/Γ with fiber V n. We recall the trivial product structure. UΩ A n is a flat A n -bundle over UΩ with a flat affine connection Ã, and UΩ V n has a flat linear connection Ṽ. The above action preserves the connections. We have a flat affine connection A on the bundle A over UΣ and a flat linear connection V on the bundle V over UΣ. We can identify bda n = S(V n ) = S n 1 where g acts by L (g) GL(n,R). We give a decomposition of Ṽ into three parts Ṽ +,Ṽ 0,Ṽ : For each vector u UΩ, we find the maximal oriented geodesic l ending at two points + l, l bdω. They correspond to the 1-dimensional vector subspaces V + ( u) and V ( u) V. Recall that bdω is C 1 since Ω is strictly convex (see [17]). There exists a unique pair of sharply supporting hyperspheres H + and H in bda n at each of + l and l. We denote by H 0 = H + H. It is a codimension 2 great sphere in bda n and corresponds to a vector subspace V 0 ( u) of codimension-two in V.

113 98 5. PROPERLY CONVEX AFFINE ACTIONS For each vector u, we find the decomposition of V as V + ( u) V 0 ( u) V ( u) and hence we can form the subbundles Ṽ +,Ṽ 0,Ṽ over UΩ where Ṽ = Ṽ + Ṽ 0 Ṽ. The map UΩ bdω by sending a vector to the endpoint of the geodesic tangent to it is C 1. The map bdω H sending a boundary point to its sharply supporting hyperspace in the space H of hyperspaces in bda n is continuous. Hence Ṽ +,Ṽ 0, and Ṽ are continuous bundles. Since the action preserves the decomposition of Ṽ, V also decomposes as (5.5) V = V + V 0 V. For each complete geodesic l in Ω, let l denote the set of unit vectors on l in one of the two directions. On l, we have a decomposition where we recall Ṽ l = Ṽ + l Ṽ 0 l Ṽ l of form l V + ( u), l V 0 ( u), l V ( u) for a vector u tangent to l V + ( u) is the space of vectors in the direction of the forward end point of l V ( u) is the space of vectors in the direction of the backward end point of l V 0 ( u) is the space vectors in directions of H 0 = H + H for l. That is, these bundles are constant bundles along l. Suppose that g Γ acts on a complete geodesic l with a unit vector u in direction of the action of g. Then V + ( u) and V ( u) corresponding to endpoints of l are eigenspaces of the largest norm λ 1 (g) of the eigenvalues and the smallest norm λ n (g) of the eigenvalues of the linear part L (g) of g. Hence (5.6) (5.7) on V + ( u), g acts by expending by λ 1 (g) λẽ(g) > 1, and on V ( u), g acts by contracting by λ n (g) λẽ(g) < 1. There exists a flow ˆΦ t : UΩ UΩ for t R given by sending v to the unit tangent vector to at α(t) where α is a geodesic tangent to v with α(0) equal to the base point of v. We define a flow on Φ t : Ã Ã by considering a unit speed geodesic flow line l in UΩ and considering l A n and acting trivially on the second factor as we go from v to ˆΦ t ( v) (See remarks in the beginning of Section 3.3 and equations in Section 4.1 of [78].) Each flow line in UΣ lifts to a flow line on A from every point in it. This induces a flow Φ t : A A. We define a flow on Φ t : V V by considering a unit speed geodesic flow line l in UΩ and and considering l V and acting trivially on the second factor as we go from v to Φ t ( v) for each t. (This generalizes the flow on [78].) Also, Φ t preserves Ṽ +, Ṽ 0, and Ṽ since on the line l, the endpoint ± l does not change. Again, this induces a flow Φ t : V V,V + V +,V 0 V 0,V V.

114 5.2. THE PROXIMAL FLOW. 99 We let S denote some metric on these bundles over UΣ/Γ defined as a fiberwise inner product: We chose a cover of Ω/Γ by compact sets K i and choosing a metric over K i A n and use the partition of unity. This induces a fiberwise metric on V as well. Pulling the metric back to à and Ṽ, we obtain a fiberwise metrics to be denoted by S. As in Section 4.4 of [78], V = V + V 0 V. By the uniform middle-eigenvalue condition, V has a fiberwise Euclidean metric S with the following properties: the flat linear connection V on V is bounded with respect to S. hyperbolicity: There exists constants C,k > 0 so that (5.8) Φ t ( v) S 1 C exp(kt) v S as t (5.9) for v V + and for v V. Φ t ( v) S C exp( kt) v S as t Using Proposition 5.4, we prove this property by taking C sufficiently large according to t 1, which is a standard technique The proof of the proximal property. We may assume that Γ has no finite order elements by taking a finite index group using Theorem Also, by Benoist [17], elements of Γ are positive bi-proximal. (See Theorem 2.28.) We can apply this to V and V + by possibly reversing the direction of the flow. The Anosov property follows from the following proposition. Let V,1 denote the subset of V of the unit length under S. PROPOSITION 5.4. Let Ω/Γ be a closed strictly convex real projective orbifold with hyperbolic fundamental group Γ. Then there exists a constant t 1 so that Φ t ( v) S C v S, v V and Φ t ( v) S C v S, v V + for t t 1 and a uniform C, 0 < C < 1. PROOF. It is sufficient to prove the first part of the inequalities since we can substitute t t and switching V + with V as the direction of the vector changed to the opposite one. Let V,1 denote the subset of V of the unit length under S. By following Lemma 5.5, the uniform convergence implies that for given 0 < ε < 1, for every vector v in V,1, there exists a uniform T so that for t > T, Φ t ( v) is in an ε-neighborhood U ε (S 0 ) of the image S 0 of the zero section. Hence, we obtain that Φ t is uniformly contracting near S 0, which implies the result. The line bundle V lifts to Ṽ where each unit vector u on Ω one associates the line V, u corresponding to the starting point in bdω of the oriented geodesic l tangent to it. Ṽ l equals l V, u. Φ t lifts to a parallel translation or constant flow Φ t of form ( u, v) ( ˆΦ t ( u), v). LEMMA 5.5. Assume Ω is strictly convex with Ω being C 1 and Γ acts properly discontinuously satisfying the uniform middle eigenvalue condition. Then Φ t V S 0 uniformly as t.

115 PROPERLY CONVEX AFFINE ACTIONS y j r y y ι l j g (l ) j j y + a FIGURE 1. The figure for Lemma 5.5. Here y i,y j denote the images under Π Ω the named points in the proof of Lemma 5.5. PROOF. Let F be a fundamental domain of UΩ under Γ. It is sufficient to prove this for Φ t on the fibers of over F of UΩ with a fiberwise metric S. We choose an arbitrary sequence {x i }, {x i } x in F. For each i, let v,i be a Euclidean unit vector in V,i := V (x i ) for the unit vector x i UΩ. That is, v,i is in the 1-dimensional subspace in R n, corresponding to the backward endpoint of the geodesic l i in Ω determined by x i in bdω and in a direction of Cl(Ω). We will show that Φ ti (x i, v,i ) S 0 for any sequence t i. This is sufficient to prove the uniform convergence to 0 by the compactness of V,1. It is sufficient to show that any sequence of {t i } has a subsequence {t j } so that Φ t j ((x i, v, j )) S 0. This follows since if the uniform convergence did not hold, then we can easily find a sequence with out such subsequences. Let y i := ˆΦ ti (x i ) for the lift of the flow ˆΦ. By construction, we recall that each Π Ω (y i ) is in the geodesic l i. Since we have the sequence of vectors x i x, x i,x F, we obtain that l i geometrically converges to a line l passing Π Ω (x) in Ω. Let y + and y be the endpoints of l where {Π Ω (y i )} y. Hence, (( v +,i )) y +,(( v,i )) y. Find a deck transformation g i so that g i (y i ) F and g i acts on the line bundle Ṽ by the linearization of the matrix of form of (5.1): (5.10) g i : V V given by (y i, v) (g i (y i ),L (g i )( v)) where 1 L (g i ) = ĥ(g λẽ(g i ) 1+ n 1 i ) : V (y i ) = V (x i ) V (g i (y i )).

116 5.2. THE PROXIMAL FLOW. 101 (Goal): We will show {(g i (y i ),L (g i )( v,i ))} 0 under S. This will complete the proof since g i acts as isometries on V with S. Also, we assume that {g i } is a sequence of mutually distinct elements since g 1 i (F) contains y i and y i forms an unbounded sequence. Since g i (l i ) F /0, we choose a subsequence of g i and relabel it g i so that {g i (l i )} converges to a nontrivial line ˆl in Ω. By our choice of l i, y i, g i as above, and Lemma 2.28, we may assume without loss of generality that each g i is positive bi-proximal since Ω is strictly convex. We choose a subsequence of {g i } so that the sequences {a i } and {r i } are convergent for the attracting fixed point a i Cl(Ω) and the repelling fixed point r i Cl(Ω) of each g i. Then {a i } a and r i r for a,r bdω. (See Figure 1.) Suppose that a = r. Then we choose an element g Γ so that g(a ) r and replace the sequence by {gg i } and replace F by F g(f). The above uniform convergence condition still holds. Then the new attracting fixed points a i g(a ) and the sequence {r i } of repelling fixed point r i of gg i converges to r also by Lemma Hence, we may assume without loss of generality that a r by replacing our sequence g i. LEMMA 5.6. Let Γ act properly discontinuously on a strictly convex domain Ω. Assume that g i is a sequence of distinct elements of Γ. Suppose that the attracting a i and repelling fixed point r i of g i form sequences converging to distinct pair of points. Then (5.11) length Ω (g i ) PROOF. Since a i a and r i r, a i r i passes a fixed compact domain U in Ω for sufficiently large i. Suppose that length Ω (g i ) < C for a constant C. Then g i (U) passes a i r i for each i. Hence, g i (U) is a subset a ball of radius 2L +C. Since {g i } form a sequence of mutually distinct elements, this contradicts the proper discontinuity of the action of Γ. Now, Lemma 5.7 shows that for every compact K Cl(Ω) {r }, (5.12) g i K {a } uniformly. Suppose that both y +,y r. Then {g i (l i )} converges to a singleton {a } by (5.12) and this cannot be since ˆl Ω. If r = y + and y bdω {r }, then g i (y i ) a by (5.12) again. Since g i (y i ) F, this is a contradiction. Therefore r = y and y + bdω {r }. Let d i = (( v +,i )) denote the other endpoint of l i from (( v,i )). Since (( v,i )) y and l i converges to a nontrivial line l, it follows that {d i = (( v +,i ))} is in a compact set in bdω {r }, and d i y +. Then {g i (d i )} a as {d i } is in a compact set in bdω {r }. Thus, {g i ((( v,i )))} y bdω where a y holds since {g i (l i )} converges to a nontrivial line ˆl in Ω as said shortly above.

117 PROPERLY CONVEX AFFINE ACTIONS Also, g i has an invariant great sphere S n 2 i bda n containing the attracting fixed point a i and sharply supporting Ω at a i. Thus, r i is uniformly bounded at a distance from S n 2 i since {r i } y = r and a i a with S n 2 i geometrically converging to a sharply supporting sphere S n 2 at a. Let E denote the standard Euclidean metric of R n. Since Π Ω (y i ) y, Π Ω (y i ) is also uniformly bounded away from a i and the tangent sphere S n 1 i at a i. Since (( v,i )) y, the vector v,i has the component v p i parallel to r i and the component v S i in the direction of S n 2 i where v,i = v p i + vs i. Since r i r = y and (( v,i )) y, we obtain v S E i 0 and that 1 C < v p i E < C for some constant C > 1. g i acts by preserving the directions of S n 2 i and r i. Since {g i ((( v,i )))} converging to y, y bdω, is bounded away from S n 2 i uniformly, we obtain that considering the homogeneous coordinates (( L (gi )( v S i ) : L (g i )( v p i ))) we obtain that the Euclidean norm of L (g i )( v S i ) L (gi )( v p i ) E is bounded above uniformly. Since r i is a repelling fixed point of g i and v p i E is uniformly bounded above, {L (g i )( v p i )} 0 by (5.3) and (5.11). {L (g i )( v p i )} 0 implies {L (g i)( v S i )} 0 for E. Hence, we obtain {L (g i )( v,i )} 0 under E. Recall that Φ t is the identity map on the second factor of UΩ V. g i ( Φ ti (x i, v,i )) = (g i (y i ),L (g i )( v,i )) F V is a vector over the compact fundamental domain F of UΩ. Since (g i (y i ),L (g i )( v,i )) is a vector over the compact fundamental domain F of UΩ with L (g i )( v,i )) E 0, we conclude that { Φ ti (x, v,i ) S } 0: For the compact fundamental domain F, the Euclidean metric E and the Riemannian metric S of Ṽ are related by a bounded constant on the compact set F. LEMMA 5.7. Let Ω be strictly convex and C 1. We choose a subsequence of {g i } so that the sequences {a i } and {r i } are convergent for the attracting fixed point a i Cl(Ω) and the repelling fixed point r i Cl(Ω) of each g i. Suppose that {a i } a and r i r for a,r bdω,a r. Suppose that g i is an unbounded sequence. Then for every compact K Cl(Ω) {r }, (5.13) g i (K) {a }

118 5.2. THE PROXIMAL FLOW. 103 uniformly. PROOF. Each g i acts on an (n 3)-dimensional subspace N gi in S n 1 disjoint from Ω. Here, N gi is the intersection of two sharply supporting hyperspaces of Ω at a i and r i. The set of {N gi } is precompact by our condition. We may assume that N gi N for an (n 2)-dimensional subspace N. Also, N gi Ω = /0. Let η i denote the complete geodesic connecting a i and r i. Let η denote the one connecting a and r. Since N is the intersection of two sharply supporting hyperspaces of Ω at a and r, η has endpoints a,r, and Ω is strictly convex, it follows η N = /0. We call P Ω for the n 2-dimensional subspace P containing N gi a slice of g i. The closure of a component of Ω with a slice of g i removed is called a half-space of g i. Let H i denote the half space of g i containing K. Since Cl(η i ) and N gi are geometrically convergent, and η N = /0, it follows that g i (P) geometrically converges to a hyperspace containing N passing a. Therefore, one deduces easily that g i (H i ) {a } geometrically. Since K H i, the lemma follows The neutralized section. We denote by Γ(V) the space of sections UΣ V and by Γ(A) the space of sections UΣ A. Recall from [78] the one parameter-group of bounded operators DΦ t, on Γ(V) and Φ t, on Γ(A). Recall Lemma 8.3 of [78] also. We denote by φ the vector field generated by this flow on UΣ. A section s : UΣ A is neutralized if (5.14) A φ s V 0. LEMMA 5.8. If ψ Γ(A), and t DΦ t, (ψ) is a path in Γ(V) that is differentiable at t = 0, then d dt (DΦ t ) (ψ) = A φ (ψ). t=0 Recall that UΣ is a recurrent set under the geodesic flow. LEMMA 5.9. A neutralized section s 0 : UΣ A exists. This lifts to a map s 0 : UΩ Ã so that s 0 γ = γ s 0 for each γ in Γ acting on à = UΩ A n. PROOF. Let s be a continuous section UΣ A. We decompose A φ (s) = A + φ (s) + A 0 φ (s) + A φ (s) V so that A ± φ (s) V ± and A 0 φ (s) V 0 hold. This can be done since along the vector field φ, V ± and V 0 are constant bundles. By the uniform convergence property of (5.8) and (5.9), the following integrals converge to smooth functions over UΣ. Again s 0 = s + 0 (DΦ t ) ( A φ (s))dt (DΦ t ) ( A + (s))dt is a continuous section and A φ (s 0) = A 0 φ (s 0) V 0 as shown in Lemma 8.4 of [78]. Since UΣ is connected, there exists a fundamental domain F so that we can lift s 0 to s 0 defined on F mapping to A. We can extend s 0 to UΩ UΩ An. 0 φ

119 PROPERLY CONVEX AFFINE ACTIONS Let N 2 (A n ) denote the space of codimension two affine subspaces of A n. We denote by G(Ω) the space of maximal oriented geodesics in Ω. We use the quotient topology on both spaces. There exists a natural action of Γ on both spaces. For each element g Γ {I}, we define N 2 (g): Now, g acts on bda n with invariant subspaces corresponding to invariant subspaces of the linear part L (g) of g. Since g and g 1 are positive proximal, a unique fixed point in bda n corresponds to the largest norm eigenvector, an attracting fixed point in bda n, and a unique fixed point in bda n corresponds to the smallest norm eigenvector, a repelling fixed point by [15] or [12]. There exists an L (g)-invariant vector subspace V 0 g complementary to the sum of the subspace generated by these eigenvectors. (This space equals V 0 ( u) for the unit tangent vector u tangent to the unique maximal geodesic l g in Ω on which g acts.) It corresponds to a g-invariant subspace M(g) of codimension two in bda n. Let c be the geodesic in UΣ that is g-invariant for g Γ. s 0 ( c) lies on a fixed affine subspace parallel to Vg 0 by the neutrality, i.e., Lemma 5.9. There exists a unique affine subspace N 2 (g) of codimension two in A n containing s 0 ( c). Immediate properties are N 2 (g) = N 2 (g m ),m Z and that g acts on N 2 (g). DEFINITION We define S (bdω) the space of hyperspaces P meeting A n where P bda n is a sharply supporting hyperspace in bda n to Ω. We denote by S(bdΩ) the space of pairs (x,h) where H S (bdω), and x is in the boundary of H and in bdω. Define to be the diagonal set of bdω bdω. Denote by Λ = bdω bdω. Let G(Ω) denote the space of maximal oriented geodesics in Ω. G(Ω) is in a one-to-one correspondence with Λ by the map taking the maximal oriented geodesic to the ordered pair of its endpoints. PROPOSITION There exists a continuous function ŝ : UΩ N 2 (A n ) equivariant with respect to Γ-actions. Given g Γ and for the unique unit speed geodesic l g in UΩ lying over a geodesic l g where g acts on, ŝ( l g ) = {N 2 (g)}. This gives a continuous map s : G ( Ω) = bdω bdω N 2 (A n ) again equivariant with respect to the Γ-actions. There exists a continuous function τ : Λ S(bdΩ). PROOF. Given a vector u UΩ, we find s 0 ( u). There exists a lift φ t : UΩ UΩ of the geodesic flow φ t. Then s 0 ( φ t ( u)) is in an affine subspace H n 2 parallel to V 0 for u by the neutrality condition (5.14). We define ŝ( u) to be this H n 2. For any unit vector u on the maximal (oriented) geodesic in Ω determined by u, we obtain ŝ( u ) = H n 2. Hence, this determines the continuous map s : G(Ω) N 2 (A n ). The Γ-equivariance comes from that of s 0. For g Γ, u and g( u) lie on the lift l g of the g-invariant geodesic l g in UΩ provided u is tangent to l g. Since g( s 0 ( u)) = s 0 (g( u)) by equivariance, g( s 0 ( u)) lies on ŝ( u) = ŝ(g( u)) in UΩ by two paragraphs above. We conclude g( s( l g)) = s( l g).

120 5.2. THE PROXIMAL FLOW. 105 The map s is defined since bdω bdω is in one-to-one correspondence with the space G(Ω). The map τ is defined by taking for each pair (x,y) Λ we take the geodesic l with endpoints x and y, and taking the hyperspace containing s(l) and its boundary containing x The asymptotic niceness. We denote by h(x,y) the hyperspace part in τ(x,y) = (x,h(x,y)). LEMMA Let U be a ΓẼ-invariant properly convex open domain in R n so that bdu bda n = Cl(Ω). Suppose that x and y are attracting and repelling fixed points of an element g of Γ in bdω. Then h(x,y) is disjoint from U. PROOF. Suppose not. h (x,y) := h(x,y) A n is a g-invariant open hemisphere, and x is an attracting fixed point of g in it. (We can choose g 1 if necessary.) Then U h(x,y) is a g-invariant properly convex open domain containing x in its boundary. Suppose first that h (x,y) has a fixed point z of g with the smallest eigenvalue in h (x,y)). Then the associated eigenvalue to z is strictly less than that of x by the uniform middle-eigenvalue condition, and hence z is in the closure of the convex open domain U h (x,y). g acts on the 2-sphere P containing x,y,z. Then g acts on P U intersecting xz o. This set P U cannot be properly convex due to the fact that z is a saddle-type fixed point. Hence, there exists no fixed point z. The alternative is as follows: h (x,y) contains a g-invariant affine subspace A of codimension at least 2 in A n, and the fixed point of the smallest eigenvalue in h (x,y) is associated with a point of bda. g h (x,y) has the largest norm eigenvalue at x,x. Therefore, we act by g on a generic point z of h (x,y) U. We obtain an arc in h (x,y) with endpoints x or x and an endpoint y in bda bda n. Here y is a fixed point in h (x,y) different from y as y h (x,y), and y Cl(U). It follows y Cl(Ω). x Cl(Ω) implies x Cl(Ω) by the proper convexity. x,y Cl(Ω) implies xy bda n Cl(Ω). Finally, xy h (x,y) for the sharply supporting subspace h (x,y) of Cl(Ω) violates the strict convexity of Ω. (See Definition 1.20 and Benoist [15].) In Theorem 7.2, we will obtain that this also gives us strict lens p-end neighborhoods. We will generalize this to Theorem 7.2. where we won t even need the existence of the properly convex domain U. LEMMA Let Γ satisfy the conditions as above. Let Γ acts on strictly convex domain Ω with Ω being C 1 in a cocompact manner. Let (x,y) bdω. Then τ(x,y) does not depend on y and is unique for each x. h (x,y) := h(x,y) A n contains s(xy) but is independent of y. h(x,y) = h(x) and h(x) is the AS-hyperspace to U. h (x,y) is never a hemisphere in bda n for every (x,y) Λ. The map τ : bdω S(bdΩ) induced from τ Ag is continuous. PROOF. We claim that for any x,y in bdω, h (x,y) is disjoint from U: By Theorem 1.1 of Benoist [15], the geodesic flow on Ω/Γ is Anosov, and hence the set of closed geodesics in Ω/Γ is dense in the space of geodesics by the basic property of the Anosov flow. Since the fixed points are in bdω, we can find a sequence x i x and y i y where x i and y i are fixed points of an element g i Γ for each i. If h (x,y) U /0, then h (x i,y i ) U /0

121 PROPERLY CONVEX AFFINE ACTIONS for i sufficiently large by the continuity of the map τ from Proposition This is a contradiction by Lemma 5.12 Also bda n does not contain h (x,y) since h (x,y) contains the s(xy) while y is chosen y x. Let H(x,y) denote the half-space in A n bounded by h (x,y) containing U. H(x,y ) is sharply supporting bdω and hence is independent of y as bdω is C 1. So, we have For each x, we define H(x,y) H(x,y ) or H(x,y) H(x,y ). H(x) := y bdω {x} H(x,y). Define h(x) as the hyperspace so that H(x) is a component of A n h(x). Clearly, h(x) is a AS-hyperspace to U. Let h (x) = h(x) A n. Now, U := x bdω H(x) contains U by the above disjointedness. Since bdω has at least n + 1 points in general position and tangent hemispheres, U is properly convex. Let U be the properly convex open domain (E Cl(H(x))). x bdω It has the boundary A (Cl(Ω)) in bda n for the antipodal map A. Since the antipodal set of bdω has at least n + 1 points in general position, U is a properly convex domain. Note that U U = /0. If for some x,y, h (x,y) is different from h (x), then h (x,y) U /0. This is a contradiction by Lemma 5.12 where U is replaced by U. Thus, we obtain h(x,y) = h(x) for all y bdω {x}. We show the continuity of x h(x): Let x i bdω be a sequence converging to x bdω. Then choose y i bdω so that y i y and we have {h(x i ) = h(x i,y i )} converges to h(x,y) = h(x) by the continuity of τ by Proposition Therefore, h is continuous. Proof of Theorem 5.3. For each point x bdω, an (n 1)-dimensional hemisphere h(x) passes A n with h(x) bda n sharply supporting Ω by Lemma Then a hemisphere H(x) A n is bounded by h(x) and contains Ω. The properly convex open domain x bdω H(x) contains U. Since bdω is C 1 and strictly convex, the uniqueness of h(x) in the proof of Lemma 5.13 gives us the unique AS-hyperspace. Let U be another properly convex open domain with the same properties as U. Then each h(x),x bdω is disjoint from U by Lemma However, U and U in the proof of Lemma 5.13 give us contradiction as in the proof. Hence, the set of AS-hyperspaces is independent of the choice of U Generalization Main argument. Now, we drop the condition of hyperbolicity on Γ. Hence, Ω, Ω bda n, is not necessarily strictly convex. Also, Ω is allowed to be the interior of a strict join. We recall some material from Section Every element of Γ is positive bi-semiproximal by Theorem 2.28 since Ω is properly convex and Γ acts on Ω cocompactly by Benoist [16]. For each bi-semi-proximal element g Γ we have two disjoint compact convex subspaces A g := A Cl(Ω) and R g := R Cl(Ω)

122 5.3. GENERALIZATION 107 for two subspaces A associated with the largest norm of eigenvalues of g and R associated with the smallest norm of the eigenvalues of g. Note g A g and g R g are both identity maps. Here, A g is associated with λ 1 (g) and R g is with λ n (g), which is an eigenvalue as well. A g is called an attracting fixed subset and R g a repelling fixed subset. By the second item of Theorem 2.28, A g is the maximal subspace where g is identity and corresponds to the eigenspaces corresponding to λ 1 (g). The multiplicity of λ 1 (g) is dima g + 1 = dim A g + 1. Also, R g is the maximal subspace where g is identity and corresponds to the eigenspaces corresponding to λ n (g). The multiplicity of λ n (g) is dimr g + 1 = dim R g + 1. Also, since g is bi-semi-proximal, there is a subspace complement to the span of A g and R g. We call this subspace the middle subspace and denote it by M g. Possibly M g = {O}. As above in (5.6) and (5.7) on A, g acts by expending by (5.15) (5.16) and on R, g acts by contracting by Also, Since Ω is convex, we obtain that λ 1 (g) λẽ(g) > 1, λ n (g) λẽ(g) < 1. A g bdω = A g, R g bdω = R g, by considering the action of g n as n ±. Here, we don t assume that Γ is not necessarily hyperbolic. hence, it is more general. Also, we obtain an asymptotically nice properly convex domain U in A n where Γ acts properly on. THEOREM We assume that Γ is a group acting on a properly convex domain Ω in bda n where Ω/Γ is a compact real projective (n 1)-orbifold. Let Γ have an action on the affine subspace A n, A n S n, acting on a properly convex domain Ω in bda n. Suppose that Ω/Γ is a closed (n 1)-dimensional orbifold, and Γ satisfies the uniform middle-eigenvalue condition. Then Γ is asymptotically nice with the properly convex open domain U, and each sharply supporting subspace Q of Ω in S n is contained in a unique AS-hyperspace to U transversal to bda n. The proof is analogous to Theorem 5.3. Now Ω is not strictly convex and hence for each point of Ω there might be more than one sharply supporting hyperspace in bda n. We generalize UΩ to the augmented unit tangent bundle (5.17) ÛΩ := { x,h a,h r ) x UΩ is a direction vector at a point of a maximal oriented geodesic l x in Ω, H a is a sharply supporting hyperspace at the starting point of l x, H r is a sharply supporting hyperspace at the ending point of l x.} Here, we regard x as a based vector and hence has an information on where it is on l and H a and H b is given orientations so that Ω is in the interior direction to them. This is not a manifold but a locally compact Hausdorff space and is a metrizable space. Since the set of sharply supporting hyperspaces of Ω at a point of bdω is compact, ÛΩ/Γ is a compact

123 PROPERLY CONVEX AFFINE ACTIONS Hausdorff space fibering over Ω/Γ with compact fibers. The obvious metric is induced from Ω and the space S n of oriented hyperspaces in S n Each element g Γ acts on a complete geodesic ending at a compact convex subspaces A g := A Cl(Ω) and R g := R Cl(Ω) for two subspaces A associated with the smallest norm of eigenvalues of g and B associated with the largest norm of the eigenvalues of g. From Section 2.5, we recall the augmented boundary bd Ag Ω. We define ˆΛ = bd Ag Ω bd Ag Ω (Π Ag Π Ag ) 1 ( ) where is the diagonal of bdω bdω. Define Ĝ(Ω) denote the set of oriented maximal geodesics in Ω with endpoints augmented with the sharply supporting hyperspace at each endpoint. The elements are called augmented geodesics. There is a one to one correspondence between ˆΛ and Ĝ(Ω). We denote by (x,h 1 )(y,h 2 ) the complete geodesic in Ω with endpoints x,y and sharply supporting hyperspaces h 1 at x and h 2 at y. For each point ( x,h a,h b ) of ÛΩ, we define V + ( x) to be the space of vectors in the direction of the forward end point of l x, V ( x) that for the backward endpoint of l x, V 0 ( x) to be the space of vectors in directions of H a H b. Now we form A by ÛΩ A n /Γ and V := ÛΩ V /Γ. For each x, V = V + ( x) V 0 ( x) V ( x). This gives us a decomposition. V = V + V 0 V. Clearly, V + and V are continuous line bundles since the beginning and the end points depends continuously on points of ÛΩ. Also, V 0 is the vector subspace of R n whose directions of nonzero vectors form H a H b. Since (H a,h b ) depends continuously on points of ÛΩ, V 0 is a continuous bundle on ÛΩ. Again a flow ˆΦ t : V V exists as above. There exists constants C,k > 0 so that (5.18) Φ t ( v) S 1 C exp(kt) v S as t for v V + and (5.19) Φ t ( v) S C exp( kt) v S as t for v V. We prove this by proving Φ t V S 0 uniformly as t i.e., Proposition 5.4 under the more general conditions that Ω is properly convex but not necessarily strictly convex. We repeat the proof here. Let F be a fundamental domain of ÛΩ under Γ. It is sufficient to prove this for Φ t on the fibers of over F of ÛΩ with a fiberwise metric S. We choose an arbitrary sequence {x i }, {x i } x in F. For each i, let v,i be a Euclidean unit vector in V,i := V (x i ) for the unit vector x i UΩ. That is, v,i is in the 1-dimensional subspace in R n, corresponding to the endpoint of the geodesic determined by x i in bdω. We generalize Lemma 5.5. We will repeat the proof since it is important to check. However, the proof is the almost same with some technical differences. LEMMA Assume that Ω is properly convex and Γ acts properly discontinuously satisfying the uniform middle eigenvalue condition with respect to bda n. Then Φ t V S 0 uniformly as t.

124 5.3. GENERALIZATION 109 R y y j yi l j g (l ) j j y + A FIGURE 2. The figure for Lemma PROOF. We proceed as in the proof of Lemma 5.5. This is sufficient to prove the uniform convergence to 0 by the compactness of V,1. (Here, (( v,i )) is an endpoint of l i in the direction given by x i.) For this, we just need to show that any sequence of {t i } has a subsequence {t j } so that Φ t j (x i, v, j ) S 0. This follows since if the uniform convergence did not hold, then we can easily find a sequence without such subsequences. Let y i := ˆΦ ti (x i ) for the lift of the flow ˆΦ. By construction, we recall that each Π Ω (y i ) is in the geodesic l i. Since we have the sequence of vectors x i x, x i,x F, we obtain that l i geometrically converges to a line l passing Π Ω (x) in Ω. Let y + and y be the endpoints of l where {Π Ω (y i )} y. Hence, (( v +,i )) y +,(( v,i )) y. (See Figure 1 for the similar situation.) Find a deck transformation g i so that g i (y i ) F, and g i acts on the line bundle Ṽ by the linearization of the matrix of form (5.1): (5.20) g i : V V given by (y i, v) (g i (y i ),L (g i )( v)) where 1 L (g i ) := ĥ(g λẽ(g i ) 1+ 1 i ) : V (y i ) = V (x i ) V (g i (y i )). n We will show {(g i (y i ),L (g i )( v,i ))} 0 under S. This will complete the proof since g i acts as isometries on V with S. Since g(y i ) F for every i, we obtain g i (l i ) F /0.

125 PROPERLY CONVEX AFFINE ACTIONS Since g i (l i ) F /0, we choose a subsequence of g i and relabel it g i so that {Π Ω (g i (l i ))} converges to a nontrivial line ˆl in Ω. We may assume that each g i is positive bi-semi-proximal by Theorem We recall facts from Section Given a generalized convergence sequence g i, we obtain an endomorphism g in M n (R) so that {((g i ))} ((g )) S(M n (R)). Recall A ({g i }) := S(Img ) Cl(Ω) and N ({g i }) := S(kerg ) Cl(Ω). We have A ({g i }),N ({g i }) bdω are both nonempty by Theorem Up to a choice of subsequence, Theorem 2.31 implies that for any compact subset K of Cl(Ω) N ({g i }) (5.21) g i (K) K A for a convex compact subset K in A. Suppose that y Cl(Ω) N ({g i }). Then g i (y i ) ŷ A ({g i }) since y i are in a compact subset of Cl(Ω) N ({g i }) and (5.21). This is a contradiction since g i (y i ) F. Hence, y N ({g i }). Let d i = (( v +,i )) denote the other endpoint of l i than (( v,i )) as above. Let d denote the limit of d i in bdω. We deduce as above up to a choice of a subsequence: Since (( v,i )) y, y N ({g i }) and l i converges to a nontrivial line l Ω and N ({g i }) is compact convex in bdω, it follows that {d i } is in a compact set in bdω N ({g i }). Then {g i (d i )} a A ({g i }) by (5.21), since {d i } is in a compact set in bdω N ({g i }). Thus, {g i ((( v,i )))} y bdω A (g i ) holds since {g i (l i )} converges to a nontrivial line in Ω. Generalizing the above argument, we may assume that y = r N ({g i }). Otherwise, we can do the same argument. Recall that E denote the standard Euclidean metric of R n. Since (( v,i )) y, the vector v,i has the component v p i parallel to N ({g i }) = ˆk i (S([m a +1,n])) and the component v S i in the orthogonal complement N ({g i }) = ˆk i (S([1,m a ])) where v,i = v p i + vs i. We may require v,i E = 1. (Here, v p i E 1 and v S i 0 since { v,i } converges to a point of N ({g i }). L (gi )( v p i ) E 0 since r gi N ({g i }) by Theorem Since {g i ((( v,i )))} converging to y, y bdω A (g i ), {g i ((( v,i )))} is uniformly bounded away from A ({g i }). Because of the orthogonal decomposition ˆk i (S([m a + 1,n]) and ˆk i (S([1,m a ])), and the fact that g i = k i d iˆk i 1, and ((L (g i ))) ((g )) in S(M n (R)), it follows that L (g i )(v S i ) either converges to zero, or (( L (g i )(v S i ))) converges to A ({g i }) by Theorem We have {L (g i )( v p i )} 0 implies {L (g i)( v S i )} 0 for E since otherwise { (( L (g i )( v S i ))) } converges to a point of A ({g i }) F ({g i }) and hence {g i ((( v,i )))} cannot be converging to y. Hence, we obtain {L (g i )( v,i )} 0 under E. Now, we find as in Section 5.1 the neutralized section s : ÛΣ A with A φ s V 0.

126 5.3. GENERALIZATION 111 Since we are looking at ÛΩ, the section s : ÛΩ N 2 (A n ), we need to look at the boundary point and a sharply supporting hyperspace at the point and find the affine subspace of dimension n 2 in R n, generalizing Proposition PROPOSITION There exists a continuous function ŝ : ÛΩ N 2 (A n ) equivariant with respect to Γ-actions. Given g Γ and for the unique unit speed geodesic l g in ÛΩ lying over an augmented geodesic l g where g acts on, ŝ( l g ) = {N 2 (g)}. This gives a continuous map s Ag : bd Ag Ω bd Ag Ω (Π Ag Π Ag ) 1 ( ) N 2 (A n ) again equivariant with respect to the Γ-actions. There exists a continuous function τ Ag : ˆΛ S(bdΩ). PROOF. The proof is entirely similar to that of Proposition but using a straightforward generalization of Lemma 5.9. We generalize Proposition We will define τ : ÛΩ S(bdΩ) as a composition of τ Ag with the map from ÛΩ to ˆΛ. This is a continuous map. Here, we don t assume that Γ acts on a properly convex domain in A n with boundary Ω. Hence, it is more general and we need a different proof. Also, we do not use the density of periodic orbits. LEMMA Let an affine group Γ acts on an affine subspace A n on a properly convex domain Ω in the boundary of an affine subspace A n. Let Γ acts on a properly convex domain Ω with a cocompact and Hausdorff quotient and satisfies the uniform middle eigenvalue condition with respect to A n. Let ((x,h 1 ),(y,h 2 ) Λ. Then τ Ag ((x,h 1 ),(y,h 2 )) does not depend on (y,h 2 ) and is unique for each (x,h 1 ). h((x,h 1 ),(y,h 2 )) contains s((x,h 1 )(y,h 2 )) but is independent of (y,h 2 ). h((x,h 1 ),(y,h 2 )) is never a hemisphere in bda n for every ((x,h 1 ),(y,h 2 )) ˆΛ. τ : bd Ag Ω S(bdΩ) is continuous. There exists an asymptotically nice convex Γ-invariant open domain U in A n with bdu A n = Ω. For every (x,h 1 ) bd Ag Ω, τ(x,h 1 ) is an AS-hyperplane of U. PROOF. Let l 1 be an augmented geodesic in Ω with end points (x,h 1 ) and (x,h 2 ) oriented towards x. Consider a connected subspace L (x,h1 ) of ÛΩ of points of maximal augmented geodesics in Ω ending at (x,h 1 ). L (x,h1 ) is in one to one correspondence with bd Ag Ω Π 1 Ag (x). We will show that τag is locally constant on L (x,h1 ) showing that it is constant. Let l 1 denote the lift of l 1 in ÛΩ. Let S be a compact neighborhood in L (x,h1 ) of a point y of l 1 transverse to l 1. The geodesic flow Φ : S R ÛΩ is distance nonincreasing on L (x,h1 ): To see this, we consider a cone containing l 1 and geodesics ending at the same points. The argument in Section of Benoist [17] shows the nonincreasing property with a small modification. Let y l 1. Consider another point y S ÛΩ with with endpoints x and x in a sharply supporting hyperplane h 2. where ((x,h 1 ),(x,h 2 )),((x,h 1 ),(x,h 2)) ˆΛ.

127 PROPERLY CONVEX AFFINE ACTIONS Let y i = Φ ti (y) l 1 be a sequence whose projection under Π Ω convergs to x. We use a deck transformation g i so that g i (y i ) F. Then g i (τ Ag (l 1 )) = τ Ag (g i (l 1 )) is a hyperspace containing g i (x) and g i (h 1 ) and ŝ(g i ( l 1 )). Let v + denote a vector in the direction of the end of l 1 other than x. Equation (5.18) shows that Φ t V + S as t. Since g i is isometry under S, and Φ ti (y) = y i and g i (y i ) F, it follows that the V + -component of g i (y i, v + ) satisfies L (g i )( v + ) S. Since g i (y i ) F and under the Euclidean norm since over a compact set F the metrics are compatible by a uniform constant, we obtain L (g i )( v + ) E. Since the affine hyperplanes in τ Ag ((x,h 1 ),(x,h 2 )) and τ Ag ((x,h 1 ),(x,h 1 )) contain x and h 1 in their boundary, they restrict to parallel hyperplanes in A. Suppose that τ Ag (((x,h 1 ),(x,h 2 )) is different from τ Ag ((x,h 1 ),(x,h 1 )) by a constant times v +. This implies that the sequence of the Euclidean distances between g i (τ Ag ((x,h 1 ),(x,h 2 ))) and g i (τ Ag ((x,h 1 ),(x,h 1))) goes to infinity as i. Now consider Φ(S [t i,t i +1]) ÛΩ, and we have obtained g i so that g i (Φ(S [t i,t i + 1])) is in a fixed compact subset ˆP of ÛΩ by the uniform boundedness of Φ(S [t i,t i +1])) shown in the second paragraph of this proof. There is a map G : UΩ ˆΛ given by sending the vector in UΩ to the ordered pair of endpoints of the geodesic passing the vector. Since ŝ is continuous, τ Ag G ˆP is uniformly bounded. The above paragraph shows that the sequence of the diameters of τ Ag G g i (Φ(S [t i,t i + 1])) can become arbitrarily large. This is a contradiction. Hence, τ Ag is constant on L (x,h1 ). This proves the first two items. The fourth item follows since τ is an induced map. The image of τ is compact since bdω is compact. This implies the third item. Define H(x,h 1 ) to be the open n-dimensional hemisphere in S n bounded by the great sphere containing the affine hyperspace τ (x,h 1 ) and containing Ω. We define U := H(x,h 1 ) A n. Then U satisfy the fifth item. (x,h 1 ) bd Ag Ω For later purposes, we again define H(x,h 1 ) to be the affine half-spaces bounded by h((x,h 1 ),(y,h 2 )) for an arbitrary choice of (y,h 2 ). PROOF OF THEOREM First, we obtain a properly convex domain where Γ acts. For each point (x,h) of bd Ag Ω, there is a hyperspace bounding a hemisphere H (x,h) containing Ω in its interior. Since bd Ag Ω is compact, the image under τ is compact. Then U := (x,h) bd Ag Ω Ho (x,h) An contains Ω. This is an open set since the compact set of H x,h, (x,h) bd Ag Ω has a lower bound on angles with bda n. Thus, U is asymptotically nice. Now, the proof is identical with that of Theorem 5.3 with Lemma 5.17 replacing Lemma Uniqueness of AS-hyperplanes. Finally, we end with some uniqueness properties. THEOREM We assume that Γ is a group acting on a properly convex domain Ω in bda n where Ω/Γ is a closed real projective (n 1)-orbifold. Let Γ have an action on the affine subspace A n, A n S n, acting on a properly convex domain Ω in bda n. Suppose that

128 5.3. GENERALIZATION 113 Ω/Γ is a closed (n 1)-dimensional orbifold and Γ satisfies the uniform middle-eigenvalue condition with respect to A n. Then if U is an asymptotically nice properly convex domain in A n where Γ acts and Cl(U) bda n = Cl(Ω), then the set of AS-planes for U containing all sharply supporting hyperspaces of Ω in bda n is independent of the choice of U. PROOF. Let U be a properly convex domain with the same properties as U stated in the premise. For each x,x bdω, and h be a supporting hyperspace of Ω in bda n. let S (x,h) be the AS-hyperspace in S n for U so that S (x,h) bda n = h. Again for each (x,h), we let S (x,h) is the AS-hyperspace in Sn for U so that S (x,h) bdan = h. Since U and U are both asymptotically nice, the sets of AS-hyperplanes are compact. For each (x,h), S (x,h) and S (x,h) differ by a uniformly bounded distance in Sn. Suppose that S (x,h1 ) is different from S (x,h 1 ) for some x,h 1 bdω. Now, we follow the argument in the proof of Lemma We again obtain a sequence g i Γ so that g i (S (x,h1 ) A n ) and g i (S(x,h 1 ) An ) are parallel affine planes and their Euclidean distances are going to as i. By compactness, we know both sequences {g i (S (x,h1 ) A n )} and {g i (S(x,h 1 ) A n )} respectively converge to two hyperplanes up to a choice of a subsequence. This means that their Euclidean distances are uniformly bounded, a contradiction. We question whether any properly convex open domain U with above properties must be always asymptotically nice. We proved this when Γ is hyperbolic in Theorem Estimating the cocycle. PROPOSITION Assume as in Theorem 5.3 or Theorem We use the notation of (5.2). Then the following holds: λẽ(g) 1/n (5.22) E b g C λ 1 (ĥ(g)) for every g Γ for a constant C,C > 0. PROOF. For each g Γ, let ĝ denote the linear part of the affine action of g in A as given by (5.2): 1 (5.23) x λẽ(g) 1+ 1 n x ĝx + ˆb g ĥ(g)x + 1 λẽ(g) b g. Let ˆΓ denote the group given by the linear action of Γ on R n. We need to estimate the size of the cocycle [ b] in H 1 (Γ,R ṋ Γ ). Recall our bundle and V := ÛΩ V /Γ where Γ acts diagonally as isometry on ÛΩ and as ĝ on V. By de Rham isomorphism theory, the cocycle [ˆb g ] corresponds to a V-valued 1-form β. (See Labourie [104] for a general theory.) For each x, V = V + ( x) V 0 ( x) V ( x). This gives us a decomposition. V = V + V 0 V. Again a flow ˆΦ t : V V exists as above which is the identity on the second factor and as a geodesic flow on UΩ. There exists constants C,k > 0 so that (5.24) for v V + and (5.25) Φ t ( v) S 1 C exp(kt) v S as t Φ t ( v) S C exp( kt) v S as t

129 PROPERLY CONVEX AFFINE ACTIONS for v V. Take a fundamental domain F of Ω. Coming back to Ṽ = ÛΩ V, we lift β to β on ÛΩ so that the Euclidean norm of β at y is bounded above by C exp(kt) where t is the distance from F to y. For any element g Γ, we take a geodesic l g from x F to g(x). We take a lift l g to UΩ. Let t g denote the length of l g. Then for the largest norm of eigenvalue λ 1 (ĝ), we have 1 C 1 expkt g λ 1 (ĝ)) C 1 expkt g for a constant C 1 > 0. That is, λ 1 (ĝ) is compatible with expkt g because of the decomposition above and how the flow acts. Also, we consider the norm of the integral l g β. The norm of the integral is bounded above by 0,t g C expktdt. Hence, we estimate ˆb g Cλ1 (ĝ) We obtain 1 λẽ(g) 1 b g C λ λẽ(g) (ĥ(g)). n REMARK Theorems 5.3 and 5.14 also generalize to the case when Γ acts on Ω as convex cocompact group: i.e., there is a convex domain C Ω so that C/Γ is compact but not necessarily closed. We work on the set of geodesics in C only and the set Λ of end points of these. In this case the limit set Λ maybe a disconnected set. The definition such as asymptotic niceness should be restricted to points of Λ only. By Proposition 5.19, we have λẽ(g i ) 1/n (5.26) E b gi C λ 1 (ĥ(g i )) for a constant C Lens type T-ends The following is another version of Theorems 5.3 and We do not assume that Γ is hyperbolic here. THEOREM Let Γ be a discrete group in SL ± (n + 1,R) (resp. PGL(n + 1,R)) acting on a properly convex domain Ω cocompactly and properly, Ω bda n S n (resp. RP n ), so that Ω/Γ is a closed n-orbifold. Suppose that Ω has a Γ-invariant open domain U forming a neighborhood of Ω in A n. Suppose that Γ satisfies the uniform middle eigenvalue condition with respect to bda n. Let P be the hyperspace containing Ω. Then Γ acts on a properly convex domain L in S n (resp. in RP n ) with strictly convex boundary L such that Ω L U, L S n P(resp. RP n P). Moreover, L is a lens-shaped neighborhood of Ω with bd L P.

130 5.4. LENS TYPE T-ENDS 115 PROOF. We prove for S n. Define a half-space H(x) A n bounded by h(x) and containing Ω in the boundary. For each H(x), x bdω, in the proofs of Theorems 5.3 and 5.14, an open n-hemisphere H (x) S n satisfies H (x) A n = H(x). Then we define V := H (x) S n x bdω is a convex open domain containing Ω as in the proof of Lemma Γ acts on a compact set H := {h h is an AS-hyperspace to U at x bdω,h S n 1 }. Let H denote the set of hemispheres bounded by an element of H and containing Ω. Then we define V := H S n H H is a convex open domain containing Ω. Here again the set of AS-hyperspaces to U is closed and bounded away from S n 1. First suppose that V is properly convex. Then V has a Γ-invariant Hilbert metric d V that is also Finsler. (See [74] and [97].) Then N ε = {x V d V (x,ω)) < ε} is a convex subset of V by Lemma 2.6. A compact tubular neighborhood M of Ω/Γ in V /Γ is diffeomorphic to Ω/Γ [ 1,1]. (See Section of [46].) We choose M in U/Γ. Since Ω is compact, the regular neighborhood has a compact closure. Thus, d V (Ω/Γ,bdM) > ε 0 for some ε 0 > 0. If ε < ε 0, then N ε M. We obtain that bdn ε /Γ is compact. Clearly, bdn/γ has two components in two respective components of (V Ω)/Γ. Let F 1 and F 2 be the fundamental domains of both components. We procure the set H j of finitely many open hemispheres H i, H i Ω, so that open sets (S n Cl(H i )) N ε cover F j for j = 1,2. By Lemma 5.23, the following is an open set containing Ω W := g(h i ) V. g Γ H i H 1 H 2 Since any path in V from Ω to bdn ε must meet bdw P first, N ε contains W and bdw. A collection of compact totally geodesic polyhedrons meet in angles < π and comprise bdw/γ. Let L be Cl(W) O. Then L has boundary only in bda n by Lemma 5.22 since Γ satisfies the uniform middle eigenvalue condition with respect to bda n. We can smooth bdw to obtain a CA-lens-neighborhood W W of Ω in N ε. Suppose that V is not properly convex. Then bdv contains v,v. V is a tube. We take any two open hemispheres S 1 and S 2 containing Cl(Ω) so that {v,v } S 1 S 2 = /0. Then g Γ g(s 1 S 2 ) V is a properly convex open domain containing Ω. and we can apply the same argument as above. [SS n ] LEMMA Let Γ be a discrete group in SL ± (n+1,r) acting on a properly convex domain Ω, Ω bda n, so that Ω/Γ is a closed n-orbifold. Suppose that Γ satisfies the uniform middle eigenvalue condition with respect to bda n and acts on a properly convex domain V in S n. Suppose that the sharply supporting hyperspheres to V at points of bdω are at uniformly bounded distances from the hypersphere containing Ω Suppose that γ i is a sequence of mutually distinct elements of Γ acting on Ω.

131 PROPERLY CONVEX AFFINE ACTIONS Let J be a compact subset of V. Then {g i (J)} can accumulate only to a subset of bdω. PROOF. Since Ω/Γ is compact, ĥ(g i ) is an unbounded sequence of elements in SL ± (n+ 1,R). Recalling (5.1), we write the elements of g i as ( 1 (5.27) ĥ(g ) λẽ (g i ) i) b 1/n gi 0 λẽ(g i ) where b gi is an n 1-vector and ĥ(g i ) is an n n-matrix of determinant ±1 and λẽ(g i ) > 0. Suppose that (( ĥ(g i ) )) ((g n 1, )) in S(M n (R)). By Proposition 5.19, we have λẽ(g i ) 1/n (5.28) E b gi C λ 1 (ĥ(g i )) for a constant C. Here, λ 1 (g i ) = λ 1(ĥ(g i )) λẽ(g i ) 1 n by the middle eigenvalue condition. Also, by Lemma 2.32, dividing by λ 1 (g i ), we have ((g i )) ((g )) where g is obtained as a limit of (5.29) 1 λ 1 (ĥ(g ))ĥ(g λẽ(gi ) i i) b λ 1 (ĥ(g i )) gi λẽ(g 0 i ) λ 1 (g i ). By the middle eigenvalue condition, we obtain that ((g i )) ((g )) where g is of form ( ) gn 1, b (5.30). 0 0 Here, b can be zero. Hence, R ({g i }) is a subset of S n 1 since the lower row is zero. Hence, g i (J) converges to g (J) S n 1. Since Γ acts properly discontinuously on Ω, we obtain g (J) bdω. LEMMA Let Γ be a discrete group of projective automorphisms of a properly convex domain V and a domain Ω V of dimension n 1. Assume that Ω/Γ is a closed n-orbifold. Suppose that Γ satisfies the uniform middle eigenvalue condition with respect to the hyperspace containing Ω. Let P be a subspace of S n so that P Cl(Ω) = /0 and P V /0. Then {g(p) V g Γ} is a locally finite collection of closed sets in V. PROOF. Suppose not. Then there exists a sequence x i g i (P) V and g i Γ so that x i x V and {g i } is a mutually distinct set of elements. We have x i F for a compact set F V. Then Lemma 5.22 applies. {g 1 i (F)} accumulates to bdω. This means that g 1 i (x i ) accumulates to bdω. Since g 1 i (x i ) P V, and P Cl(Ω) = /0, this is a contradiction. REMARK In [78], the authors uses the term recurrent geodesic. A geodesic is recurrent if it accumulates to compact subsets in both directions. They work in a compact subsurface where geodesics are recurrent in both directions. In our work, since Ω/Γ is a closed orbifold, every geodesic is recurrent in their sense. Hence, their theory generalizes here.

132 CHAPTER 6 Properly convex radial ends and totally geodesic ends: lens properties We will consider properly convex ends in this chapter. In Section 6.1, we define the uniform middle eigenvalue conditions for R-ends and T-ends. We state the main results of this chapter the equivalence of these conditions with the generalized lens conditions for R-ends or T-ends. The generalized lens conditions often improve to lens conditions. In Section 6.2, we start to study the R-end theory. First, we discuss the holonomy representation spaces. Tubular actions and the dual theory of affine actions are discussed. We show that distanced actions and asymptotically nice actions are dual. Hence, the uniform middle eigenvalue condition implies the existence of the distanced action by using the dual theory in Chapter 5. In Section 6.3, we show that the uniform middle-eigenvalue condition of a properly convex end is equivalent to the lens-shaped property of the end under some assumptions. In particular, this is true for virtually factorable properly convex ends. First, we estimate the largest eigenvalue λ 1 (g) in terms of word length. Next, we study orbits under the action with the uniform middle eigenvalue conditions. We show how to make a strictly convex boundary of a lens. We now prove the main results Theorems 6.6 and 6.7. In Section 6.6, we discuss the properties of lens-shaped ends. We show that if the holonomy is strongly irreducible, the lens shaped ends have concave neighborhoods. If the generalized lens-shaped end is virtually factorable, then it can be made into a lens-shaped totally-geodesic R-end, which is a surprising result. In Section 6.7, we discuss the theory of lens-shaped T-ends. The theory basically follows from that of lens-shaped R-ends. We obtain the duality between the lens-shaped T-ends and generalized lens-shaped R-ends. The main reason we are studying the lens-shaped R-ends are to use them in studying the deformations preserving the convexity properties. These objects are useful in trying to understand this phenomenon. We also remark, sometimes, a lens-shaped p-end neighborhood may not exist for an R-p-end. However, generalized lens-shaped p-end neighborhood may exists for the R-p-end. REMARK 6.1. There is an independent approach to the end theory by Cooper, Long, Leitner, and Tillmann announced in the summer of Our theory overlaps with theirs in many cases. (See [108], [107], and [109].) However, their ends have nilpotent fundamental groups. They approach gives us some what simpler criterions to tell the existence of these types of ends. REMARK 6.2. The results in this Chapter will be proved for S n -versions. However, the RP n -version of the results are implies by the S n -version Main results. DEFINITION 6.3. Suppose that Ẽ is a R-p-end of generalized lens-type. Then Ẽ have a p-end-neighborhood that is projectively diffeomorphic to the interior of {p} L o {p} 117

133 PROPERLY CONVEX R-ENDS AND T-ENDS under dev where {p} L is a generalized lens-cone over a generalized lens L where ({p} L {p}) = + L, and let h(π 1 (Ẽ) acts on L properly and cocompactly. A concave pseudoend-neighborhood of Ẽ is the open pseudo-end-neighborhood in O projectively diffeomorphic to {p} L {p} L Uniform middle eigenvalue conditions. The following applies to both R-ends and T-ends. Let Ẽ be a p-end and ΓẼ the associated p-end holonomy group. We say that Ẽ is non-virtually-factorable if any finite index subgroup has a finite center or ΓẼ is virtually center-free; otherwise, Ẽ is virtually factorable. (See Section ) Let ΣẼ denote the universal cover of the end orbifold ΣẼ associated with Ẽ. We recall Proposition 2.49 (Theorem 1.1 of Benoist [18]). If ΓẼ is virtually factorable, then ΓẼ satisfies the following condition: Cl( ΣẼ) = K 1 K k where each K i is properly convex or is a singleton. Let G i be the restriction of the K i -stabilizing subgroup of ΓẼ to K i. Then G i acts on K o i cocompactly. (Here K i can be a singleton and Γ i a trivial group. ) A finite index subgroup G of ΓẼ is isomorphic to a cocompact subgroup of Z k 1 G 1 G k where G i is the restriction to K i of the subgroup of ΓẼ acting on K i. The center Z k 1 of G is a subgroup acting trivially on each K i. Note that there are examples of discrete groups of form ΓẼ where G i are non-discrete. (See also Example of [121] as pointed out by M. Kapovich.) We will use Z k 1 to simply represent the corresponding group on ΓẼ. Here, Z k 1 is called a virtual center of ΓẼ. Let Γ be generated by finitely many elements g 1,...,g m. Let w(g) denote the minimum word length of g G written as words of g 1,...,g m. The conjugate word length cwl(g) of g π 1 (Ẽ) is min{w(cgc 1 ) c π 1 (Ẽ)}. Let d K denote the Hilbert metric of the interior K o of a properly convex domain K in RP n or S n. Suppose that a projective automorphism group Γ acts on K properly. Let length K (g) denote the infimum of {d K (x,g(x)) x K o }, compatible with cwl(g). DEFINITION 6.4. Let vẽ be a p-end vertex of an R-p-end Ẽ. Let K := Cl( ΣẼ). The p-end holonomy group ΓẼ satisfies the uniform middle eigenvalue condition with respect to vẽ or the R-p-end structure if the following hold: each g ΓẼ satisfies for a uniform C > 1 independent of g ( ) λ(g) (6.1) C 1 length K (g) log Clength λ vẽ (g) K (g), for λ(g) equal to the largest norm of the eigenvalues of g and the eigenvalue λ vẽ (g) of g at vẽ. (Of course we choose the matrix of g so that λ vẽ (g) > 0. See Remark 1.1 as we are looking for the lifting of g that acts on p-end neighborhood.) We can replace length K (g) with cwl(g) for properly convex ends. We remark that the condition does depend on the choice of vẽ; however, the radial end structures will determine the end vertices. The definition of course applies to the case when ΓẼ has the finite-index subgroup with the above properties.

134 6.1. MAIN RESULTS. 119 We recall a dual definition identical with the definition in Section 5.1 but adopted to T-p-ends. DEFINITION 6.5. Suppose that Ẽ is a properly convex T-p-end. Suppose that the ideal boundary component ΣẼ of the T-p-end is properly convex. Let K = Cl( ΣẼ). Let g : R n+1 R n+1 be the dual transformation of g : R n+1 R n+1. The p-end holonomy group ΓẼ satisfies the uniform middle-eigenvalue condition with respect to ΣẼ or the T-pend structure if it satisfies if each g ΓẼ satisfies for a uniform C > 1 independent of g ( ) λ(g) (6.2) C 1 length K (g) log λ K (g Clength ) K (g), for the largest norm λ(g) of the eigenvalues of g for the eigenvalue λ K (g ) of g in the vector in the direction of K, the point dual to the hyperspace containing K. Again, the condition depends on the choice of the hyperspace containing ΣẼ, i.e., the T-p-end structure. (We again lift g so that λ K (g) > 0.) Here ΓẼ will act on a properly convex domain K o of lower dimension, and we will apply the definition here. This condition is similar to ones studied by Guichard and Wienhard [84], and the results also seem similar. We do not use their theories. Our main tools to understand these questions are in Chapter 5 which we will use here. We will see that the condition is an open condition; and hence a structurally stable one. (See Corollary 7.3.) Main results. As holonomy groups, the conditions for being a generalized lens R-p-end and being a lens R-p-end are equivalent. For the following, we are not concerned with a lens-cone being in O. THEOREM 6.6 (Lens holonomy). Let Ẽ be an R-p-end of a strongly tame convex real projective orbifold. Then the holonomy group h(π 1 (Ẽ)) satisfies the uniform middle eigenvalue condition for the R-p-end vertex vẽ if and only if it acts on a lens-cone and its lens properly and cocompactly. Moreover, in this case, the lens-cone exists in the union of convex segments with the vertex vẽ in the directions of the region Ω S n 1 vẽ where h(π 1 (Ẽ)) acts properly discontinuously. For the following, we are concerned with a lens-cone being in O. THEOREM 6.7 (Actual lens-cone). Let O be a strongly tame convex real projective orbifold. Let Ẽ be a properly convex R-p-end. The p-end holonomy group satisfies the uniform middle-eigenvalue condition if and only if Ẽ is a generalized lens-shaped R-p-end. Assume that the holonomy group of O is strongly irreducible and O is properly convex. If O satisfies the triangle condition (see Definition 6.25) or Ẽ is virtually factorable or is a totally geodesic R-end, then we can replace the term generalized lens-shaped to lens-shaped in the above statement. This is repeated as Theorem We will prove the analogous result for totally geodesic ends in Theorem Notice that there is no condition on O to be properly convex.

135 PROPERLY CONVEX R-ENDS AND T-ENDS Another main result is on the duality of lens-shaped ends: Let RP n = P(R n+1 ) be the dual real projective space of RP n. In Section 6.2, we define the projective dual domain Ω in RP n to a properly convex domain Ω in RP n where the dual group Γ to Γ acts on. Vinberg showed that there is a duality diffeomorphism between Ω/Γ and Ω /Γ. The ends of O and O are in a one-to-one correspondence. Horospherical ends are dual to themselves, i.e., self-dual types, and properly convex R-ends and T-ends are dual to one another. (See Proposition 6.34.) We will see that generalized lens-shaped properly convex R-ends are always dual to lens-shaped T-ends by Corollary The end theory In this section, we discuss the properties of lens-shaped radial and totally geodesic ends and their duality also The holonomy homomorphisms of the end fundamental groups: the tubes. We will discuss for S n only here but the obvious RP n -version exists for the theory. Let Ẽ be an R-p-end of O. Let SL ± (n + 1,R) vẽ be the subgroup of SL ± (n + 1,R) fixing a point vẽ S n. This group can be understood as follows by letting vẽ = [0,...,0,1] as a group of matrices: For g SL ± (n + 1,R) vẽ, we have ( ) 1 (6.3) λ vẽ (g) 1/n ĥ(g) 0 v g λ vẽ (g) where ĥ(g) SL ± (n,r), v R n,λ vẽ (g) R +, is the so-called linear part of h. Here, λ vẽ : g λ vẽ (g) for g SL ± (n + 1,R) vẽ is a homomorphism so it is trivial in the commutator group [ΓẼ, ΓẼ]. There is a group homomorphism (6.4) L : SL ± (n + 1,R) vẽ SL ± (n,r) R + g (ĥ(g),λ vẽ (g)) with the kernel equal to R n, a dual space to R n. Thus, we obtain a diffeomorphism We note the multiplication rules SL ± (n + 1,R) vẽ SL ± (n,r) R n R +. (6.5) (A, v,λ)(b, w, µ) = (AB, 1 µ 1/n vb + λ w,λ µ). We denote by L 1 : SL ± (n + 1,R) vẽ SL ± (n,r) the further projection to SL ± (n,r). Let ΣẼ be the end (n 1)-orbifold. Given a representation ĥ : π 1 (ΣẼ) SL ± (n,r) and a homomorphism λ vẽ : π 1 (ΣẼ) R +, we denote by R ṋ the R-module with the π h,λ 1 (ΣẼ)-action given by vẽ g v = 1 ĥ(g)( v). λ vẽ (g) 1/n And we denote by R n the dual vector space with the right dual action given by ĥ,λ vẽ g v = 1 λ vẽ (g) 1/n ĥ(g) ( v).

136 6.2. THE END THEORY 121 Let H 1 (π 1 (Ẽ),R n ) denote the cohomology space of 1-cocycles v(g) R n. ĥ,λ vẽ ĥ,λ vẽ As Hom(π 1 (ΣẼ),R + ) equals H 1 (π 1 (ΣẼ),R), we obtain: THEOREM 6.8. Let O be a strongly tame properly convex real projective orbifold, and let O be its universal cover. Let ΣẼ be the end orbifold associated with an R-p-end Ẽ of O. Then the space of representations is the fiber space over Hom(π 1 (ΣẼ),SL ± (n + 1,R) vẽ )/SL ± (n + 1,R) vẽ Hom(π 1 (ΣẼ),SL ± (n,r))/sl ± (n,r) H 1 (π 1 (ΣẼ),R) with the fiber isomorphic to H 1 (π 1 (ΣẼ),R n ) for each ([ĥ],λ). ĥ,λ vẽ Over the stable part of Hom(π 1 (ΣẼ),SL ± (n,r))/sl ± (n,r), the dimensions of the fibers are constant (see Johnson-Millson [90]). A similar idea is given by Mess [118]. In fact, the dualizing these matrices gives us a representation to Aff(A n ). (See Chapter 5.) In particular if we restrict ourselves to linear parts to be in SO(n,1), then we are exactly in the cases studied by Mess. (The concept of the duality is explained in Section 6.2.3) Tubular actions. Let us give a pair of antipodal points v and v. If a group Γ of projective automorphisms fixes a pair of fixed points v and v, then Γ is said to be tubular. There is a projection Π v : S n {v,v } Sv n 1 given by sending every great segment with endpoints v and v to the sphere of directions at v. f A tube in S n (resp. in RP n ) is the closure of the inverse image Π 1 v (Ω) of a convex domain Ω in S n 1 v (resp. in RP n 1 v ). We denote the closure in S n by T v, which we call a tube domain. Given an R-p-end Ẽ of O, let v := vẽ. The end domain is R v ( O). If an R-p-end Ẽ has the end domain ΣẼ = R v ( O), h(π 1 (Ẽ)) acts on T v. The image of the tube domain T v in RP n is still called a tube domain and denoted by T v where v is the image of v. We will now discuss for the S n -version but the RP n version is obviously clearly obtained from this by a minor modification. Letting v have the coordinates [0,...,0,1], we obtain the matrix of g of π 1 (Ẽ) of form ( 1 ) ĥ(g) 0 (6.6) λ v (g) 1 n b g λ v (g) where b g is an n 1-vector and ĥ(g) is an n n-matrix of determinant ±1 and λ v (g) is a positive constant. Note that the representation ĥ : π 1 (Ẽ) SL ± (n,r) is given by g ĥ(g). Here we have λ v (g) > 0. If ΣẼ is properly convex, then the convex tubular domain and the action is said to be properly tubular Affine actions dual to tubular actions. Let S n 1 in S n = S(R n+1 ) be a great sphere of dimension n 1. A component of a component of the complement of S n 1 can be identified with an affine space A n. The subgroup of projective automorphisms preserving S n 1 and the components equals the affine group Aff(A n ). By duality, a great (n 1)-sphere S n 1 corresponds to a point v S n 1. Thus, for a group Γ in Aff(A n ), the dual groups Γ acts on S n := S(R n+1, ) fixing v S n 1. (See Proposition 2.58 also.)

137 PROPERLY CONVEX R-ENDS AND T-ENDS A hyperspace of S m for 0 m n, supports a convex domain Ω if it passes bdω but disjoint from Ω o. An oriented hypersurface S m for 0 m n, supports a convex domain Ω if the hypersurface supports Ω and the open hemisphere bounded by it in the orientation direction contains Ω o. Let S n 1 denote a hyperspace in S n. Suppose that Γ acts on a properly convex open domain U where Ω := bdu S n 1 is a properly convex domain. We recall that Γ has a properly convex affine action. Let us recall some facts from Section 2.5. A great (n 2)-sphere P S n is dual to a great circle P in S n given as the set of hyperspheres containing P. The great sphere S n 1 S n with an orientation is dual to a point v S n and it with an opposite orientation is dual to v S n. An oriented hyperspace P S n 1 of dimension n 2 is dual to an oriented great circle passing v and v, giving us an element P of the linking sphere Sv n 1 of rays from v in S n. The space S of oriented hyperspaces in S n 1 equals S n 1. Thus, there is a projective isomorphism I 2 : S = S n 1 P P S n 1 v For the following, let s use the terminology that an oriented hyperspace V in S i supports an open submanifold A if it bounds an open i-hemisphere H in the right orientation containing A. PROPOSITION 6.9. Suppose that Γ SL ± (n + 1,R) acts on a properly convex open domain Ω S n 1 cocompactly. Then the dual group Γ acts on a properly tubular domain B with vertices v := v S n 1 and v := v S n 1, dual to Sn 1. The domain Ω and domain R v (B) in the linking sphere S n 1 v from v in the directions of B o are projectively diffeomorphic to a pair of dual domains in S n 1 respectively. PROOF. Given Ω S n 1, we obtain the properly convex open dual domain Ω in S n 1. An oriented n 2-hemisphere sharply supporting Ω in S n 1 corresponds to a point of bdω o and vice versa. (See Section 2.5.) An oriented great n 1-sphere in S n sharply supporting Ω but not containing contains a great n 2-sphere P in S n 1 sharply supporting Ω. The dual P of P is the set of hyperspaces containing P, a great circle in S n. The set of oriented great n 1-spheres containing P sharply supporting Ω but not containing Ω forms a pencil; in this case, a great open segment I P in S n with endpoints v and v. Let P S n 1 v denote the dual of P in terms of S n 1. Then P := I 2 (P ) is the direction of P at v as we can see from the projective isomorphism I 2. Now P supports Ω if and only if P Ω. Hence, there is a homeomorphism I P := {Q Q is an oriented great n 1-sphere sharply supporting Ω,Q S n 1 = P} S P = {p p is a point of a great open segment in P with endpoints v,v (6.7) where the direction P = I 2 (P ),P Ω }. The set B of oriented hyperspaces sharply supporting Ω but not containing Ω meets an oriented (n 2)-hyperspace in S n 1 sharply supporting Ω. Thus, we obtain B = S P S n. P Ω

138 6.2. THE END THEORY 123 Let T (Ω ) denote the union of open great segments with endpoints v and v in direction of Ω. Thus, B = T (Ω ). Thus, there is a homeomorphism (6.8) I := {Q Q is an oriented great n 1-sphere sharply supporting Ω} S = {p p S P,P bdω } = bdb {v,v }. Also, R v (B ) = Ω by B := T (Ω ). Thus, Γ acts on Ω if and only if Γ acts on I if and only if Γ acts on S if and only if Γ acts on B and on Ω. Since these are properly convex open domains, and the actions are cocompact, they are uniquely determined up to projective diffeomorphisms Distanced tubular actions and asymptotically nice affine actions. We remark that there is an approach to this by D. Fried for representations with linear parts in SO(2, 1) using cocycles and stability alternative to the approach of this section [69]. The approach is similar to what we did in Chapter 5 but is in the dual setting. Hence, we can think of our approach as a generalization of this work. DEFINITION Radial action: A properly tubular action of Γ is said to be distanced if a Γ-invariant tubular domain contains a properly convex compact Γ-invariant subset disjoint from the vertices of the tubes. Affine action: We recall from A properly convex affine action of Γ is said to be asymptotically nice if Γ acts on a properly convex open domain U in A n with boundary in Ω S n 1, and Γ acts on a compact subset J := {H H is an AS-hyperspace at x bdω,h S n 1 } where we require that every sharply supporting (n 2)-dimensional space of Ω in S n 1 is contained in at least one of the element of J. The following is a simple consequence of the homeomorphism given by equation (6.8). PROPOSITION Let Γ and Γ be dual groups where Γ has an affine action on A n and Γ is tubular with the vertex v = v S n 1 dual to the boundary S n 1 of A n. Let Γ = (Γ ) acts on a convex open domain Ω with a closed n-orbifold Ω/Γ. Then Γ acts asymptotically nicely if and only if Γ acts on a properly tubular domain B and is distanced. THEOREM Let Γ be a nontrivial properly convex tubular action at vertex v = v S n 1 on S n (resp. in RP n ) and acts on a properly convex tube B and satisfies the uniform middle-eigenvalue conditions with respect to v S n 1. We assume that Γ acts on a convex open domain Ω S n 1 v where B = T (Ω) and Ω/Γ is a closed n-orbifold. Then Γ is distanced inside the tube B. Furthermore, K meets each open boundary great segment in B at a unique point. Finally, K is contained in a hypersphere disjoint from v,v when Γ is virtually factorable. PROOF. Let v be the vertex of B. Let Ω denote the convex domain in S n 1 v corresponding to B o. By Theorems 5.3 and 5.14, Γ is asymptotically nice. Proposition 6.11 implies the result. Now, we prove the final part to show the total geodesic property of virtually factorable ends: Suppose that Γ acts virtually reducibly on Sv n 1 on a properly convex domain Ω. Then Γ is virtually isomorphic to a cocompact subgroup of Z l 0 1 Γ 1 Γ l0

139 PROPERLY CONVEX R-ENDS AND T-ENDS where Γ i is irreducible. by Proposition 2.49 ([16]). Also, Γ acts on K := K 1 K l0 = Cl(Ω) S n 1 v where K i denotes the properly convex compact set in S n 1 v where Γ i acts on for each i. Γ i acts trivially on K j for j i. Here, K i is 0-dimensional for i = 1,...,s. Let B i be the convex tube with vertices v and v corresponding to K i. Each Γ i for i = 1,...,s acts on a nontrivial tube B i with vertices v and v in a subspace. For each i, s + 1 i r, B i is a great segment with endpoints v and v. A point p i corresponds to B i in S n 1 v. The virtual center isomorphic to Z l 0 1 is in the group ΓẼ. Recall that a nontrivial element g of the virtual center acts trivially on the subspace K i of Sv n 1 ; that is, g has only one associated eigenvalue in points of K i. There exists a nontrivial element g of the virtual center with the largest norm eigenvalue in K i since the action of ΓẼ on ΣẼ is compact. By the middle eigenvalue condition, for each i, we can find g in the center so that g has a hyperspace K i B i with largest norm eigenvalues. Since Γ i acts on K i and commutes with g, Γ i also acts on K i. The convex hull of K 1 K l 0 in Cl(B) is a distanced Γ-invariant compact convex set. For (ζ 1,,ζ l0 ) R l 0 +, we define (6.9) ζ 1 I n ζ 2 I n ζ (ζ 1,,ζ l0 ) :=......,ζ n ζ n ζ n l 0 +1 l 0 = 1, 0 0 ζ l0 I nl0 +1 in the coordinates where K i corresponds to the blocks. Now, we consider the general case. The element x of K o S n 1 has coordinates Ẽ (λ 1,...,λ l,x 1,...,x l ), λ i = 1, where x = l i=1 λ ix i for x i is a unit vector in the direction of Ki o. Let Γ denote the finite index normal subgroup acting on each of K i in ΓẼ. We take the Zariski closure of Γ. It is isomorphic to R l 0 1 Z (Γ 1 ) Z (Γ l0 ) where Z (Γ i ) is the Zariski closure of Γ i easily derivable from Theorem 1.1 of Benoist [16] for our setting. The elements of R l 0 commutes with elements of Γ i and hence with Γ. For any element k R l 0, we define ζ (exp( k))) as the matrix with diagonal entries exp(k 1 ),,exp(k l0 ). Then there is a linear map Z : Z l0 1 R l 0 so that an isomorphism Z l0 1 Γ is represented by ζ exp Z. Let logλ 1 : Z l0 1 R denote map given by taking the log of the largest norm and logλ n : Z l0 1 R given by taking the log of the smallest norm and logλ vẽ : Z l0 1 R the log of the eigenvalue at vẽ. Now, logλ 1 and logλ n extends to piecewise linear function on R l0 1 that are linear over cones with origin as the vertex. logλ 1 has only nonnegative values and logλ 1 has nonpositive values. The uniform middle eigenvalue condition is equivalent to the condition that logλ 1 > logλ vẽ > logλ n holds over R n {O}.

140 6.3. THE CHARACTERIZATION OF LENS-SHAPED REPRESENTATIONS 125 Let B i denote the tube T (K i ). We choose an element g of the center having largest norm of the eigenvalue at K i as an automorphism of S n 1 vẽ. g acts on B i. By the uniform middle eigenvalue conditions, g fixes an subspace ˆK i equal to B i P i for a hyperspace P i in the span of B i corresponding to the largest norm eigenvalue of g as an element of SL ± (n + 1,R). By commutativity, the center also acts on ˆK i. Hence, the center acts on the join ˆK 1 ˆK l0 equals T P for a hyperspace P. By commutativity, Γ acts on B i also. Suppose that for some g ΓẼ Γ, g(p) P. Then g(p) B j has a point x closer to v or v than P B j for some j. Assume that it is closer to v without loss of generality. We find a sequence { k i } so that g i = ζ exp Z( k i ) have the largest eigenvalue at B i and λ 1 (g i )/λ vẽ (g i ). Since λ n (g 1 ) = λ 1 (g i ) 1 and λ vẽ (g 1 i ) = λ vẽ (g i ) 1, we obtain that g 1 i (x) v as i. Then we obtain that g i (g(p)) T is not distanced. This contradicts the first paragraph of the proof The characterization of lens-shaped representations The main purpose of this section is to characterize the lens-shaped representations in terms of eigenvalues, a major result of this monograph. First, we prove the eigenvalue estimation in terms of lengths for non-virtually-factorable and hyperbolic ends. We show that the uniform middle-eigenvalue conditions imply the existence of limits. This proves Theorem 6.6. Finally, we prove the equivalence of the lens condition and the uniform middle-eigenvalue condition in Theorem 6.26 for both R-ends and T-ends under very general conditions. That is, we prove Theorem 6.7. Techniques here are somewhat related to the work of Guichard-Wienhard [84] and Benoist [13]. Also, when the linear part is in SO(2,1), D. Fried has proven similar results without going to the dual space using cocycle conditions. We do not present it here The eigenvalue estimations. Let O be a properly convex real projective orbifold and O be the universal cover in S n. Let Ẽ be a properly convex R-p-end of O, and let vẽ be the p-end vertex. Let h : π 1 (Ẽ) SL ± (n + 1,R) vẽ be a homomorphism and suppose that π 1 (Ẽ) is hyperbolic. In this article, we assume that h satisfies the middle eigenvalue condition. We denote by the norms of eigenvalues of g by (6.10) λ 1 (g),...,λ n (g),λ vẽ (g),where λ 1 (g) λ n (g)λ vẽ (g) = ±1, and λ 1 (g)... λ n (g), where we allow repetitions. Recall the linear part homomorphism L 1 from the beginning of Section 6.2. We denote by ĥ : π 1 (Ẽ) SL ± (n,r) the homomorphism L 1 h. Since ĥ is a holonomy of a closed convex real projective (n 1)-orbifold, and ΣẼ is assumed to be properly convex, ĥ(π 1 (Ẽ)) divides a properly convex domain ΣẼ in S n 1 vẽ. We denote by λ 1 (g),..., λ n (g) the norms of eigenvalues of ĥ(g) so that (6.11) λ1 (g)... λ n (g), λ 1 (g)... λ n (g) = ±1 hold. These are called the relative norms of eigenvalues of g. We have λ i (g) = λ i (g)/λ vẽ (g) 1/n for i = 1,..,n.

141 PROPERLY CONVEX R-ENDS AND T-ENDS Note here that eigenvalues corresponding to λ 1 (g), λ 1 (g),λ n (g), λ n (g),λ vẽ (g) are all positive since the nonidentity elements are semi-proximal by Benoist [17]. We define ( ) ( ) λ1 (g) λ1 (g) length(g) := log = log. λ n (g) λ n (g) This equals the infimum of the Hilbert metric lengths of the associated closed curves in ΣẼ/ĥ(π 1 (Ẽ)) as first shown by Kuiper. (See [12] for example.) We recall the notions in Section (See [12] and [13] also.) When Γ acts on a properly convex domain cocompactly and properly then every nonelliptic elements are positive bi-semiproximal by Theorem (See [18] also). Since ΣẼ is properly convex, all infinite order elements of ĥ(π 1 (Ẽ)) are positive bi-semiproximal and a finite index subgroup has only positive bi-semiproximal elements and the identity. When π 1 (Ẽ) is hyperbolic, all infinite order elements of ĥ(π 1 (Ẽ)) are positive biproximal and a finite index subgroup has only positive biproximal elements and the identity. Assume that ΓẼ is hyperbolic. Suppose that g ΓẼ is proximal. We define (6.12) α g := log λ 1 (g) log λ n (g) log λ 1 (g) log λ n 1 (g),β g := log λ 1 (g) log λ n (g) log λ 1 (g) log λ 2 (g), and denote by Γ p the set of proximal elements. We define Ẽ β ΓẼ := sup β g,α ΓẼ := inf α g. g Γ p g Γ p Ẽ Ẽ Proposition 20 of Guichard [83] shows that we have (6.13) 1 < α ΣẼ α Γ 2 β Γ β ΣẼ < for constants α ΣẼ and β ΣẼ depending only on ΣẼ since ΣẼ is properly and strictly convex. Here, it follows that α ΓẼ,β ΓẼ depends on ĥ, and they form positive-valued functions on the union of components of Hom(π 1 (Ẽ),SL ± (n + 1,R))/SL ± (n + 1,R) consisting of convex divisible representations with the algebraic convergence topology as given by Benoist [18]. THEOREM Let O be a strongly tame convex real projective orbifold. Let Ẽ be a properly convex R-p-end of the universal cover O, O S n, n 2. Let ΓẼ be a hyperbolic group. Then ( ) ( ) n 2 length(g) log n β λ 1 (g) n 2 length(g) ΓẼ n α ΓẼ for every nonelliptic element g ĥ(π 1 (Ẽ)). PROOF. Since there is a positive bi-proximal subgroup of finite index, we concentrate on positive bi-proximal elements only. We obtain from above that log λ 1 (g) λ n (g) log λ 1 (g) λ 2 (g) β ΓẼ.

142 6.3. THE CHARACTERIZATION OF LENS-SHAPED REPRESENTATIONS 127 We deduce that (6.14) λ 1 (g) λ 2 (g) ( λ1 (g) λ n (g) Since we have λ i λ 2 for i 2, we obtain ) ( ) 1/βΩ 1/βΓẼ λ1 (g) = = exp( λ n (g) ) length(g). β ΓẼ (6.15) ( ) λ 1 (g) 1/βΓẼ λ i (g) λ1 λ n and since λ 1 λ n = 1, we have We obtain λ 1 (g) n = λ 1 (g) λ 2 (g) λ1 (g) λ 1 (g) λ n 1 (g) λ n (g) (6.16) log λ 1 (g) 1 n By similar reasoning, we also obtain (6.17) log λ 1 (g) 1 n ( ( 1 + n 2 β ΓẼ 1 + n 2 α ΓẼ ) ) ( λ1 (g) λ n (g) length(g). length(g). ) n 2 +1 β ΓẼ. REMARK Under the assumption of Theorem 6.13, if we do not assume that π 1 (Ẽ) is hyperbolic, then we obtain (6.18) 1 n length(g) log λ 1 (g) n 1 n length(g) for every semiproximal element g ĥ(π 1 (Ẽ)). PROOF. Let λ i (g) denote the norms of ĥ(g) for i = 1,2,...,n. hold. We deduce log λ 1 (g)... log λ n (g),log λ 1 (g) + + log λ n (g) = 0 log λ n (g) = logλ 1 log λ n 1 (g) (6.19) ( n 1 (n 1)log λ 1 log λ 1 (g) 1 n 1 log λ n (g) ) log λ 1 (g) log λ 1 (g) 1 n 1 log λ 1 (g) λ n (g) 1 n length(g).

143 PROPERLY CONVEX R-ENDS AND T-ENDS We also deduce (6.20) log λ 1 (g) =log λ 2 (g) + + log λ n (g) (n 1)log λ n (g) (n 1)log λ 1 (g) λ n (g) (n 1)log λ n (g) log λ 1 (g) nlog λ 1 (g) n 1 n length(g) log λ 1 (g). REMARK We cannot show that the middle-eigenvalue condition implies the uniform middle-eigenvalue condition. This could be false. For example, we could obtain a sequence of elements g i Γ so that λ 1 (g i )/λ vẽ (g i ) 1 while Γ satisfies the middleeigenvalue condition. Certainly, we could have an element g where λ 1 (g) = λ vẽ (g). However, even if there is no such element, we might still have a counter-example. For example, suppose that we might have ( ) λ log 1 (g i ) λ vẽ (g i ) 0. length(g) (If the orbifold were to be homotopy-equivalent to the end orbifold, this could happen by changing λ vẽ considered as a homomorphism π 1 (ΣẼ) R +. Such assignments are not really understood globally but see Benoist [12]. Also, an analogous phenomenon seems to happen with the Margulis space-time and diffused Margulis invariants as investigated by Charette, Drumm, Goldman, Labourie, and Margulis recently. See [78]) The uniform middle-eigenvalue conditions and the orbits. Let Ẽ be a properly convex R-p-end of the universal cover O of a strongly tame properly convex real projective orbifold O. Assume that ΓẼ satisfies the uniform middle-eigenvalue condition. There exists a ΓẼ-invariant convex set to be denoted ΛẼ distanced from {vẽ,vẽ } by Theorem For the corresponding tube T vẽ, ΛẼ bdt vẽ is a compact subset distanced from {vẽ,vẽ }. Let C H (ΛẼ) be the convex hull of ΛẼ in the tube TẼ obtained by Theorem Then C H (ΛẼ) is a ΓẼ-invariant distanced subset of T vẽ. We may assume that C H (ΛẼ) bdt vẽ = ΛẼ by replacing ΛẼ if necessary. Also, ΛẼ bdt vẽ contains all attracting and repelling fixed points of γ ΓẼ by invariance and the middle-eigenvalue condition. Recall Definition 2.1 of Chapter 4 on geometric limits Hyperbolic groups. We first consider when ΓẼ is hyperbolic. LEMMA Let O be a strongly tame convex real projective orbifold. Let Ẽ be a properly convex R-p-end. Assume that ΓẼ is hyperbolic and satisfies the uniform middle eigenvalue conditions. Suppose that γ i is a sequence of elements of ΓẼ acting on T vẽ. The sequence of attracting fixed points a i and the sequence of repelling fixed points b i are so that a i a and b i b where a,b are not in {vẽ,vẽ }. Suppose that the sequence {λ i } of eigenvalues where λ i corresponds to a i converges to +.

144 6.3. THE CHARACTERIZATION OF LENS-SHAPED REPRESENTATIONS 129 Let M := T vẽ Cl( b i vẽ b i vẽ ). i=1 Then the point a is in the compact subset Λ in T vẽ {vẽ,vẽ } that is the geometric limit of {γ i (K)} for any compact subset K M. PROOF. We may assume without loss of generality that a b since otherwise we replace {g i } with {gg i } where g(a ) b. Proving for this case implies the general cases. Let k i be the inverse of the factor min { λ1 (γ i ) λ 2 (γ i ), λ1 (γ i ) λ vẽ (γ i ) n+1 n } = λ 1(γ i ). λ vẽ (γ i ) Then k i 0 by the uniform middle eigenvalue condition and (6.14). There exists a totally geodesic sphere S n 1 i at b i sharply supporting T vẽ. a i is uniformly bounded away from S n 1 i for i sufficiently large. S n 1 i bounds an open hemisphere H i containing a i where a i is the attracting fixed point by Corollary of [95] or by Proposition 2.20 so that for a Euclidean metric d E,i, γ i H i : H i H i we have (6.21) d E,i (γ i (x),γ i (y)) k i d E,i (x,y),x,y H i. Note that {Cl(H i )} converges geometrically to Cl(H) for an open hemisphere containing a in the interior. Actually, we can choose a Euclidean metric d E,i on H o i so that {d E,i J J} is uniformly convergent for any compact subset J of H. Hence there exists a uniform positive constant C so that (6.22) d(a i,k) < C d Ei (a i,k). provided a i,k J and sufficiently large i. Since ΓẼ is hyperbolic, the domain Ω corresponding to T vẽ in S n 1 vẽ is strictly convex. For any compact subset K of M, the equation K M is equivalent to K Cl( b i vẽ b i vẽ ) = /0. i=1 Since the boundary sphere bdh meets Cl(T vẽ ) in this set only by the strict convexity of Ω, we obtain K bdh = /0. And K H since Cl(T vẽ ) Cl(H ). We have d(k,bdh ) > ε 0 for ε 0 > 0. Thus, the distance d(k,bdh i ) is uniformly bounded by a constant δ. d(k,bdh i ) > δ implies that d Ei (a i,k) C/δ for a positive constant C > 0 Acting by g i, we obtain d Ei (g i (K),a i ) k i C/δ by (6.21), which implies d(g i (K),a i ) C k i C/δ by (6.22). Since {k i } 0, the fact that {a i } a implies that {g i (K)} geometrically converges to a. LEMMA Let O be a strongly tame convex real projective orbifold. Let Ẽ be a properly convex R-p-end. Assume that ΓẼ is hyperbolic, and satisfies the uniform middle eigenvalue conditions. Suppose that γ i is a convergence sequence of elements of ΓẼ acting on T vẽ. Let M := T o vẽ Then ΛẼ contains the geometric limit of any subsequence of {γ i (K)} for any compact subset K M. Furthermore, Â ({γ i }) Cl(T vẽ ),R ({γ i }) Cl(T vẽ ) ΛẼ.

145 PROPERLY CONVEX R-ENDS AND T-ENDS PROOF. Let z T o vẽ {vẽ,vẽ }. Let [z] denote the corresponding element in ΣẼ. Let {γ i } be any sequence in ΓẼ so that the corresponding sequence {γ i ([z])} in ΣẼ S n 1 vẽ converges to a point z in bdσẽ S n 1 vẽ. Clearly, a fixed point of g ΓẼ {I} in bdt vẽ {vẽ,vẽ } is in ΛẼ since g has at most one fixed point on each open segment in the boundary. For the attracting fixed points a i and r i of γ i, we can assume that {a i } a,{r i } r for a i,r i ΛẼ where a,r ΛẼ by the closedness of ΛẼ. Assume a r first. By Lemma 6.16, we have {γ i (z)} a and hence the limit z = a. However, it could be that a = r. In this case, we choose γ 0 ΓẼ so that γ 0 (a) r. Then γ 0 γ i has the attracting fixed point a i so that we obtain {a i } γ 0(a) and repelling fixed points r i so that {r i } r holds by Lemma Then as above {γ 0 γ i (z)} γ 0 (a) and we need to multiply by γ0 1 now to show {γ i (z)} a. Thus, the limit set is contained in ΛẼ. An attracting fixed point of g ΓẼ must be in ΛẼ since ΛẼ is ΓẼ-invariant. The set of attracting fixed point of g in Cl( ΣẼ) S n 1 is dense by [17]. LEMMA Let {g i } be a sequence of projective automorphisms acting on a strictly convex domain Ω in S n (resp. RP n ). Suppose that the sequence of attracting fixed points {a i bdω} a and the sequence of repelling fixed points {r i bdω} r. Assume that the corresponding sequence of eigenvalues of a i limits to + and that of r i limits to 0. Let g be any projective automorphism of Ω. Then {gg i } has the sequence of attracting fixed points {a i } converging to g(a) and the sequence of repelling fixed points converging to r. PROOF. Recall that g is a quasi-isometry. Given ε > 0 and a compact ball B disjoint from a ball around r, we obtain that gg i (B) is in a ball of radius ε of g(a) for sufficiently large i. For a choice of B and a sufficiently large i, we obtain gg i (B) B o. Since gg i (B) B o, we obtain (gg i ) n (B) (gg i ) m (B) o for n > m by induction, There exists an attracting fixed point a i of gg i in gg i (B). Since the diameter of gg i (B) is converging to 0, we obtain that {a i } g(a). Also, given ε > 0 and a compact ball B disjoint from a ball around g(a), g 1 i g 1 (B) is in the ball of radius ε of r. Similarly to above, we obtain the needed conclusion Non-hyperbolic groups. Now, we generalize to not necessarily hyperbolic ΓẼ. Since the ΓẼ-invariant distanced set ΛẼ contains the attracting fixed set A i and the repelling fixed set R i of any g ΓẼ. Hence, their limits A,B are subset of ΛẼ. We denote by ˆR i the intersection of the γ i -invariant hyperspace sharply supporting T vẽ and containing R i. LEMMA Let O be a strongly tame convex real projective orbifold. Let Ẽ be a properly convex R-p-end. Assume that ΓẼ is non-hyperbolic and or virtually-factorable and satisfies the uniform middle eigenvalue conditions with respect to vẽ. Suppose that γ i is a generalized convergence sequence of elements of ΓẼ acting on T vẽ. Then ΛẼ contains the geometric limit of any subsequence of {γ i (K)} for any compact subset K T o vẽ. Furthermore, Â ({γ i }) Cl(T vẽ ),R ({γ i }) Cl(T vẽ ) ΛẼ.

146 6.3. THE CHARACTERIZATION OF LENS-SHAPED REPRESENTATIONS 131 PROOF. Let L = C H (ΛẼ) M. Then L is a convex set uniformly bounded away from vẽ and its antipode. by a geometric consideration. Let ΩẼ denote the set of directions of segments from vẽ in T vẽ. Since ΩẼ is strictly convex, a geometric consideration shows that Cl(L) bdt vẽ = ΛẼ. Otherwise, ΩẼ cannot be strictly convex. Given any sequence g i, we can extract a convergence sequence g i with a convergence limit g. Suppose that L o = /0. Then L is a convex domain on a hyperspace P disjoint from vẽ. We use a coordinate system where each γ Γ is of form (6.6) where b g = 0. Dividing g i by λ 1 (g i ) and taking a limit, we obtain that g equals ( ) ĝ 0 (6.23) 0 0 by the uniform middle eigenvalue condition and Lemma Hence A ({g i }) P. By Theorem 2.37, A ({g i }) P Cl(T vẽ ) = ΛẼ. The remainders are simple to show. Suppose that L o is not empty. Then L N ({g i }) = /0 by Lemma Given any convergence sequence g i,g i Γ converging to g, elements of g i (x) for x L converges to a point of A ({g i }). By Lemma 2.32, vẽ N ({g i }) since λ vẽ (g i )/λ(g i ) 0 by the uniform middle eigenvalue condition. Dividing g i by λ 1 (g i ) and taking a limit, we obtain that g equals ( ) ĝ 0 (6.24) ˆb 0 by the uniform middle eigenvalue condition and Lemma 2.32 and Proposition 5.19 by dualizing the proof of Lemma Here ĝ is not zero as in the proof of Lemma Since ĝ 0, the image of g is now a subspace of the same dimension as A ({ĝ i }). Actually, it is graph over A ({ĝ i }) where the vertical direction is given by the direction to vẽ for a linear function given by ˆb. Since ΓẼ acts on L, g (x) Cl(L) bdt vẽ. Hence, g (L) = A ({g i }) ΛẼ. Using }, we obtain R ({g i }) ΛẼ.  ({g i }) does not contain vẽ or its antipode.  ({g i }) Cl(T vẽ ) maps homeomorphic to  ({ĝ i }) Cl(Ω) = A ({ĝ i }) by projecting. By the above description of g by (6.24), we will show that A ({g i }) =  ({g i }) Cl(T vẽ ): Any element of x T vẽ satisfies x = [ v x ], v x = v L + c vẽ for a constant c > 0 and a vector v L in direction of L and a vector vẽ in direction of vẽ. Then g ( v x ) = g ( v L ) + cg ( vẽ). {g 1 i Since cg ( vẽ) = 0, we obtained that g (T vẽ ) = g (L). Since g (T vẽ ) =  ({g i }) Cl(T vẽ ), and g (L) = A ({g i }), we obtain the result. The final statement is also proved here. For the following, ΓẼ can be virtually factorable. PROPOSITION Let O be a strongly tame convex real projective orbifold. Let Ẽ be a properly convex R-p-end. Assume that ΓẼ satisfies the uniform middle eigenvalue condition with respect to the R-p-end structure. Let vẽ be the R-end vertex and z T o vẽ. Then there exists a ΓẼ-invariant distanced compact convex set ˆL in Cl(T vẽ ) {vẽ,vẽ } satisfying the following properties :

147 PROPERLY CONVEX R-ENDS AND T-ENDS (i) ΛẼ = ˆL T vẽ and contains the limit points of orbits of each compact subset of T vẽ. ΛẼ contains all attracting fixed sets of elements of ΓẼ. If ΓẼ is hyperbolic, then the set of attracting fixed point is dense in the set. (ii) For each segment s in T vẽ with an endpoint vẽ, the great segment containing s meets ΛẼ at a unique point other than vẽ,vẽ. That is, there is a one-to-one correspondence between bdσẽ and ΛẼ. Furthermore, for any ΓẼ-distanced set L in Cl(L ) Cl(T vẽ ) {vẽ,vẽ } we have ΛẼ = L T vẽ. (iii) ΛẼ is homeomorphic to S n 2. PROOF. Let ΛẼ be any given ΓẼ-invariant distanced compact set in Cl(T vẽ ) {vẽ,vẽ } by Theorem (A) Consider first when ΓẼ is not virtually factorable and hyperbolic. Proposition 6.17 proves (i) here: Let L be the closure of C H (ΛẼ), which is ΓẼ-invariant. Let K = L bdcl(t vẽ ) {vẽ,vẽ }. Clearly ΛẼ K. Since ΓẼ is hyperbolic, any point y of bd ΣẼ S n 1 vẽ is a limit point of some sequence {g i (x)} for x ΣẼ by [17]. Thus, at least one point in the segment l y containing y with endpoints vẽ and vẽ is a limit point of some subsequence of {g i (x)} by Lemma Thus, l y ΛẼ /0. and l y K /0. Let us choose a convenient Euclidean metric E for R n+1. Also, l y K is a unique: Suppose not. Let z and z be the two points. We choose a line l in ΣẼ ending at y. Let y i be the sequence of points on l covering to y. We choose g i as in the proof of Lemma 5.5 so that g i (y) F for a compact fundamental domain F. The radial segment vẽy corresponds to a point of Cl(ΣẼ), which corresponds to a unit vector v in R n under the projection R n {O} S n 1 vẽ. Then as in the proof of Lemma 5.5 (6.25) 1 ĥ(g λẽ(g i ) 1+ 1 i )( v ) 0 n in the Euclidean metric writing λ vẽ (g i ) = λẽ(g i ). Let v E denote the unit vector in the direction of vẽ. We consider R n to be a complementary subspace to this vector under the norm. The vector for z can be written as v z = λ v + v E, and the vector for z can be written v z = λ v + v E. Then by (6.3). Let us denote g i ( v z ) = λ 1 ĥ(g λ 1 i )( v ) + (λ b gi v + λ vẽ (g i )) v E ñ E (g i) c i := 1 ĥ(g λ 1 i )( v ) ñ E (g. i) E Since the direction of g i ( v z ) is bounded away from vẽ, λ b gi v + λ vẽ (g i ) c i c i is uniformly bounded. By (6.25), λ vẽ (g i ). c i

148 6.3. THE CHARACTERIZATION OF LENS-SHAPED REPRESENTATIONS 133 Hence, b gi v as i. c i We also have g i ( v z ) = λ 1 ĥ(g λ 1 i )( v ) + (λ b gi v + λ vẽ (g i )) v E. ñ E (g i) Since λ λ, and b gi v λ b gi v c i c i + λ vẽ (g i ) v E c i = (λ λ) b gi v c i + λ b gi v c i + λ vẽ (g i ) v E c i cannot be uniformly bounded. This implies that g i (z ) converges to vẽ or vẽ. Since z ΛẼ and ΛẼ is ΓẼ, this is a contradiction. Thus, K = ΛẼ, and (i) and (ii) hold for ΛẼ. For the last part of (ii), suppose that we have another distanced set L. We take a convex hull of L L and apply the same reasoning as above. (iii) Since ΛẼ is closed and compact and bounded away from vẽ,vẽ, the section s : bdωẽ bdtẽ is continuous. If not, we can contradict (ii). (B) Now suppose that ΓẼ is not virtually factorable and is not hyperbolic. Lemma 6.19 prove that the orbits limit to ΛẼ only. An attracting fixed sets in TẼ is in ΛẼ as in case (A). First suppose that a maximal segment η in TẼ corresponds to an element y of bd ΣẼ. Now we take a line l in ΣẼ as in the hyperbolic case. Then h(g i )( v ) 0 as above using Lemma 5.15 instead of Lemma 5.5. The identical argument will show that η o meets with ΛẼ at a unique point. This proves (i) and (ii). (iii) follows as above. (C) Suppose that Γ E is virtually factorable. We follow the proof of Theorem Now, T vẽ corresponds to a properly convex domain Ω that is the interior of the strict join K 1 K l. Then a totally geodesic ΓẼ-invariant hyperspace H is disjoint from {vẽ,vẽ } and meets O by the proof of Theorem Here, we may regard K i H for each i = 1,...,l. Then consider any sequence g i so that g i (x) x 0 for a point x T o vẽ and x 0 T vẽ. Let x denote the corresponding point of ΣẼ for x. Then g i (x ) converges to a point y S n 1 vẽ. Let x R n+1 be the vector in the direction of x. We write x = x E + x H where x H is in the direction of H and x E is in the direction of vẽ. By the uniform middle eigenvalue condition and estimating the size of vectors, we obtain g i (x) x 0 for x ΛẼ and x 0 H. Hence, x 0 H ΛẼ. Thus, every limit point of an orbit of x is in H. If there is a point y in ΛẼ H, then there is a strict join K i1 K il vẽ for a proper collection containing y. As in the proof of Theorem 6.12, by Proposition 2.49, we can find a sequence g i, virtually central g i ΓẼ, so that g i K i1 K il converges to the identity and the maximal norm λ 1 (g i ) of the eigenvalue associated with K i1 K il and λẽ(g i ) satisfy λ 1 (g i )/λẽ(g i ) 0 by the uniform middle eigenvalue condition choosing the maximal norm of g i to be in the complementary domains. Hence g i (y) vẽ. Again this is a contradiction. Hence, we obtain ΛẼ H. (i), (ii), (iii) follow easily now.

149 PROPERLY CONVEX R-ENDS AND T-ENDS 6.4. Convex cocompact actions of the p-end holonomy groups. DEFINITION A (resp. generalized) lens-shaped R-p-end with the p-end vertex vẽ is strictly (resp. generalized) lens-shaped if we can choose a (resp. generalized) CAlens domain D with the top hypersurfaces A and the bottom one B so that each great open segment in S n from vẽ in the direction of bd ΣẼ meets Cl(D) A B at a unique point. A (resp. generalized) lens L is called strict lens if the following hold: Cl(A) = Cl(A) A = Cl(B) = Cl(B) B, A B = L, and Cl(A) Cl(B) = Cl(L). Recall that in order that L is to be a lens, we assume that π 1 (Ẽ) acts cocompactly on L. Also, this set must equal ΛẼ since Cl(A) A is the set of limit points of ΓẼ of the fundamental domain of A. Obviously, a lens of a lens-shaped R-p-end is strict if and only if the R-p-end is strictly lens-shaped. In this section, we will prove Proposition 6.22 obtaining a lens. PROPOSITION Let O be a strongly tame convex real projective orbifold. Assume that the universal cover O is a subset of S n. Let ΓẼ be the holonomy group of a properly convex R-p-end Ẽ. Let T vẽ be an open tube corresponding to R(vẼ). Suppose that ΓẼ satisfies the uniform middle eigenvalue condition with respect to the R-p-end structure, and acts on a distanced compact convex set L in Cl(T vẽ ) where L T vẽ lifts under dev to O as an embedded subset. Then any open p-end-neighborhood U containing a lift to O of L T vẽ contains a lenscone p-end-neighborhood of the R-p-end Ẽ. Furthermore, every lens of the cone is a strict lens. PROOF. We may assume that U embeds to a neighborhood of L under a developing map by taking U sufficiently small. We denote by U the image again. By assumption, U L has two components since either ΓẼ acts a totally geodesic hyperspace meeting the rays from vẽ transversally, or L o T vẽ /0 and L T vẽ has two boundary components closer and farther away from vẽ. Let ΛẼ denote bdt vẽ L. Let us choose finitely many points z 1,...,z m U L in the two components of U L. Proposition 6.20 shows that the orbits of z i for each i accumulate to points of ΛẼ only. Hence, a totally geodesic hypersphere separates vẽ with these orbit points and another one separates vẽ and the orbit points. Define the convex hull C 2 := C H (ΓẼ({z 1,...,z m }) L). Thus, C 2 is a compact convex set disjoint from vẽ and vẽ and C 2 bdt vẽ = ΛẼ. (See Definition 2.14.) We need the following lemma. LEMMA We continue to assume as above. Then we can choose z 1,...,z m in U so that for C 2 := C H (ΓẼ({z 1,...,z m }) L), bdc 2 O is disjoint from L and C 2 U. PROOF. First, suppose L o /0. Then (bdl T vẽ )/ΓẼ is diffeomorphic to a disjoint union of two copies of ΣẼ. We can cover a compact fundamental domain of bdl T vẽ by the interior of n-balls in O that are convex hulls of finite sets of points in U. Since (L O)/ΓẼ is compact, there exists a positive lower bound of {d (x,bdu) x L}. Let F O

150 6.4. CONVEX COCOMPACT ACTIONS OF THE P-END HOLONOMY GROUPS. 135 denote the union of these finite sets. We can choose ε > 0 so that the ε-do -neighborhood U of L in O is a subset of U. Moreover U is convex by Lemma 2.6 following [51]. The convex hull C 2 is a union of simplices with vertices in ΓẼ(F). If we choose F to be in U, then C 2 is in U by convexity as well. The disjointedness of bdc 2 from L T vẽ follows since the ΓẼ-orbits of above balls cover bdl T vẽ. If L o = /0, then K is in a hyperspace. The reasoning is similar to the above. We continue: LEMMA Let L be as above. Let C be a ΓẼ-invariant distanced compact convex set with boundary in where (C T õ E )/ΓẼ is compact. There are two components A and B of bdc T õ meeting every great segment in T õ. Suppose that A (resp. B ) are disjoint E E from L. Then A (resp. B ) contains no line ending in bdo. PROOF. It is enough to prove for A. Suppose that there exists a line l in A ending at a point of bdt vẽ. Assume l A. The line l projects to a line l in Ẽ. Let C 1 = C T vẽ. Since A/ΓẼ and B/ΓẼ are both compact, and there exists a fibration C 1 /ΓẼ A/ΓẼ induced from C 1 A using the foliation by great segments with endpoints vẽ,vẽ. Since A/ΓẼ is compact, we choose a compact fundamental domain F in A and choose a sequence {x i l} whose image sequence in l converges to the endpoint of l in bd ΣẼ. We choose γ i Γ vẽ so that γ i (x i ) F where {γ i (Cl(l ))} geometrically converges to a segment l with both endpoints in bd ΣẼ. Hence, {γ i (Cl(l))} geometrically converges to a segment l in A. We can assume that for the endpoint z of l in A, γ i (z) converges to the endpoint p 1. Proposition 6.20 implies that the endpoint p 1 of l is in LẼ := L TẼ. Let t be the endpoint of l not equal to z. Then t A. Since γ i is not a bounded sequence, γ i (t) converges to a point of ΛẼ. Thus, both endpoints of l are in ΛẼ and hence l o L by the convexity of L. However, l A implies that l o A. As A is disjoint from L, this is a contradiction. The similar conclusion holds for B. Since A and analogously B do not contain any geodesic ending at bdo, bdc 1 bdt vẽ is a union of compact n 1-dimensional simplices meeting one another in strictly convex dihedral angles. By choosing {z 1,...,z m } sufficiently close to bdc 1, we may assume that bdc 1 bdt vẽ is in O. Now by smoothing bdc 1 bdt vẽ, we obtain two boundary components of a lens. Let F denote the compact fundamental domain of the boundary of the lens. The strictness of the lens follows from Proposition 6.20 since the boundary of the lens is a union of orbits of F and the limit points are only in ΛẼ. This completes the proof of Proposition Proof of Theorem 6.6. First, we show that the uniform middle eigenvalue condition implies the existence of CA-lens: Let T vẽ denote the tube domain with vertices vẽ and vẽ. Let ΛẼ denote the intersection of bdt vẽ with the distanced compact ΓẼ-invariant convex set L by Theorem Let C 1 be the convex hull of L and the ΓẼ-orbits of finite number of points in the inner component of T vẽ L so that bdc 1 T vẽ is disjoint from L. By Lemma 6.24, the component bdc 1 T vẽ contains no line l with endpoints x,y in L, and hence can be isotopied to be strictly convex and smooth as above. Also, L/Γ is a compact orbifold since we added a finite number of balls to L up to the Γ-action. Hence, L is a CA-lens.

151 PROPERLY CONVEX R-ENDS AND T-ENDS Now, we show the converse. Let L be a CA-lens of the lens-cone where ΓẼ acts cocompactly on. Let T vẽ be the tube corresponding to L. Let g ΓẼ be an infinite order element. Then g is bi-semi-proximal by Theorem Suppose that λ vẽ (g) > λ 1 (g) for any Then g n (x), x C must accumulate to vẽ or vẽ, which contradicts the disjoint of Cl(L) to vẽ and vẽ. If λ vẽ (g) = λ 1 (g), then let l g be the line in ΣẼ where g acts on. Let P g be the 2-dimensional subspace where g acts on. Then g acts on C P g. Since it is a strictly convex arc, g cannot act on it with the eigenvalue condition. Therefore, ΓẼ satisfies the middle-eigenvalue condition that λ 1 (g)/λ vẽ (g) > 1 for every infinite order g. There is a map ΓẼ H 1 (ΓẼ,R) obtained by taking a homology class. The above map g logλ vẽ (g) induces homomorphism Λ h : H 1 (ΓẼ,R) R that depends on the holonomy homomorphism h. Suppose that ΓẼ does not satisfy the uniform middle-eigenvalue condition with respect to vẽ. Then there exists a sequence of elements g i so that ( log λ 1 (g i ) λ vẽ (g i ) length(g i ) ) 0 as i. Note that we can change h by only changing the homomorphism Λ h and still obtain a representation. Let [g ] denote a limit point of {[g i ]/length(g i )} in the space of currents on ΣẼ. We can make a sufficiently small change of h so that Λ h ([g ]) < 0 From this, we obtain that (6.26) log ( λ 1 (g i ) λ h vẽ (g i ) ) < 0 for some g i Γ. This changes the action of ΓẼ. Proposition 7.4 implies that the perturbed CA-lens L is still a properly convex domain with the same tube domain. We know that T vẽ is not changed, and a small perturbation of boundary components of a CA-lens of the lens-shaped end remains strictly convex and transversal to great segments in T vẽ in a compact fundamental domain of L. The changed ΓẼ-action gives us the strict convexity and the transversality to the leaves of the foliation of the perturbed boundary components. We may consider a compact neighborhood of a fundamental domain of L. Perturbation can be understood at this compact set. In particular, the CA-lens perturbed by a sufficiently small amount is distanced since a sharply supporting hyperspace at a point of a boundary component of L is disjoint from vẽ and vẽ by the transversality to great segments. By (6.26), we obtain that λ 1 (g) < λ h vẽ (g) for some g for the largest eigenvalue λ 1 (g) of h(g) and that λ h vẽ (g) at vẽ. This implies as above Cl(L) contains vẽ or vẽ since we can consider the sequence g i (L) as i. This is a contradiction since the CA-lens L is disjoint from it. To show the final statements, notice that we just need the premises there to prove the forward direction.

152 6.5. THE UNIFORM MIDDLE-EIGENVALUE CONDITIONS AND THE LENS-SHAPED ENDS The uniform middle-eigenvalue conditions and the lens-shaped ends. Now, we aim to prove Theorem 6.7 restated as Theorem DEFINITION We say that a strongly tame properly convex O with O S n (resp. RP n ) satisfies the triangle condition if for any fixed end-neighborhood system of O, every triangle T Cl( O), if T bdo,t o O, and T Cl(U) /0 for a radial p-end neighborhood U, then T is a subset of Cl(U) bdo. By Corollary 7.19, strongly tame strict SPC-orbifolds with generalized lens-shaped or horospherical ends satisfy this condition. The converse is not necessarily true. A minimal ΓẼ-invariant distanced compact set is the smallest compact ΓẼ-invariant distanced set in TẼ. THEOREM Let O be a strongly tame convex real projective orbifold. Let ΓẼ be the holonomy group of a properly convex R-end Ẽ and the end vertex vẽ. Then the following are equivalent: (i) Ẽ is a generalized lens-shaped R-end. (ii) ΓẼ satisfies the uniform middle-eigenvalue condition with respect to vẽ. Assume now π 1 (O) is strongly irreducible and O is properly convex. If O furthermore satisfies the triangle condition or, alternatively, assume that Ẽ is virtually factorable, then the following holds: ΓẼ is lens-shaped if and only if ΓẼ satisfies the uniform middle-eigenvalue condition. PROOF. Again, we will prove first for S n. Thus, assume O S n. (ii) (i): This follows from Theorem 6.6 since we can intersect the lens with O to obtain a generalized lens and generalized lens-cone from it. (Here, of course π 1 (Ẽ) acts cocompactly on the generalized lens.) (i) (ii): Let L be a generalized CA-lens in the generalized lens cone L vẽ. Let B be the lower boundary component of L in the tube TẼ. Since B is strictly convex, the upper component of TẼ B is a properly convex domain, which we denote by U. Let l x denote the maximal segment from vẽ passing x for x U L. We define a function f : U L R given by f (x) to be the Hilbert distance on the line l x from x to L l x. Then a level set of f is always strictly convex: This follows by taking a 2-plane P containing vẽ passing L. Let x,y be a points of f 1 (c) for a constant c > 0. Let x be the point of Cl(L) l x closest to x and y be one of Cl(L) l y closest to y. Let x be one of Cl(L) l x furthest from x. Let y be one of Cl(L) l y furthest from x. Since f (x) = f (y), a cross-ratio argument shows that the lines extending xy,x y and x y are concurrent outside U P. The strict convexity of B shows that f (z) < ε for z xy o. We can approximate each level set by a polyhedral hypersurface in U L convex at vertices using the convexity of the level set. Then we can smooth it to be a strictly convex hypersurface. Let V denote the domain bounded by this and B. Then V has a strictly convex smooth boundary in U. Theorem 6.6 implies (ii). The final part follows by Lemma LEMMA Suppose that O is a strongly tame properly convex real projective orbifold and satisfies the triangle condition or, alternatively, assume that an R-p-end Ẽ is virtually factorable. Suppose that the holonomy group Γ is strongly irreducible. Then the R-p-end Ẽ is generalized lens-shaped if and only if it is lens-shaped.

153 PROPERLY CONVEX R-ENDS AND T-ENDS PROOF. If Ẽ is virtually factorable, this follows by Theorem 6.31 (iv). Suppose that Ẽ is not virtually factorable. Now assume the triangle condition. Thus, given a generalized CA-lens L, let L b denote Cl(L) Cl(T vẽ ). We obtain the convex hull M of L b. M is a subset of Cl(L). The lower boundary component of L is a smooth convex surface. Let M 1 be the outer component of bdm T vẽ. Suppose that M 1 meets bdo. M 1 is a union of the interior of simplices. By Lemma 2.46, a simplex σ in Cl( O) is either in bdo or its interior σ o is disjoint from it. Hence, there is a simplex σ in M 1 bdo. Taking the convex hull of vẽ and an edge in σ, we obtain a triangle T with T bdo and T o O. This contradicts the triangle condition by Lemma Thus, M 1 O. By Theorem 6.26, the end satisfies the uniform middle eigenvalue condition. By Proposition 6.22, we obtain a lens-cone in O. LEMMA Suppose that O is a strongly tame properly convex real projective orbifold and satisfies the triangle condition. Then no triangle T with T o O, T bdo has a vertex equal to an R-p-end vertex. PROOF. Let vẽ be a p-end vertex. Choose a fixed radially foliated p-end-neighborhood system. Suppose that a triangle T with T bdo contains a vertex equal to a p-end vertex. Let U be an inverse image of a radially foliated end-neighborhood in the end-neighborhood system, and be a p-end neighborhood of a p-end Ẽ with a p-end vertex vẽ. By the triangle condition, T Cl(U) bdo. Since U is foliated by radial lines from vẽ, we choose U so that bdu O covers a compact hypersurface in O. Let U denote the set of segments in Cl(U) from vẽ. Every segment in U in the direction of ΣẼ ends in bdu O. Also, the segments U in directions of bd ΣẼ are in bdu bdo by the definition of ΣẼ. Also, Cl(U) is a union of segments in U. Thus, Cl(U) bdo is a union of segments in directions of bd ΣẼ. Since T o O, each segment in U with interior in T o is not in directions of bd ΣẼ. Let w be the end point of the maximal extension in O of such a segment. Then w is not in Cl(U) bdo by the conclusion of the above paragraph. This contradicts T Cl(U) bdo The properties of lens-shaped ends. A trivial one-dimensional cone is an open half-space in R 1 given by x > 0 or x < 0. LEMMA Let O be a strongly tame properly convex orbifold. Then the d O - diameter of the boundary of a concave end-neighborhood of an R-end E is bounded by the Hilbert diameter of the end orbifold Σ E of E. Also, given any end-neighborhood, there is a concave end-neighborhood in it. PROOF. Let Ẽ be a p-end corresponding to E. Let U be a concave p-end neighborhood of Ẽ that is a cone: U is the interior of {v} L L for a generalized CA-lens L and the p-end vertex v corresponding to U. Then B := U O is a smooth lower boundary component of L. Any maximal segment in O tangent to B at x must end in bdt for the tube T for U since the segment is in the sharply supporting hyperspace of L at x. Thus, there is a projection Π : B ΣẼ that is a diffeomorphism. Moreover this is a Finsler isometry by considering the Finsler metric restricted to the tangent space to B at x to that of the tangent space to ΣẼ at Π(x). The conclusion follows.

154 6.6. THE PROPERTIES OF LENS-SHAPED ENDS. 139 Suppose that we have a lens-cone V that is a p-end-neighborhood equal to the interior of L vẽ where L is a generalized CA-lens bounded away from vẽ. Now take a p-end neighborhood U. We assume without loss of generality that U covers a product end-neighborhood with compact boundary. By taking smaller U if necessary, we may assume that U and L are disjoint. Since bdu /h(π 1 (Ẽ)) and L/h(π 1 (Ẽ)) are compact, ε > 0. Let L := {x V d V (x,l) ε}. Since a lower component of L is strictly convex, we can show that L can be polyhedrally approximated and smoothed to be a generalized CA-lens by Lemma 2.6. Clearly, h(π 1 (Ẽ)) acts on L. We choose sufficiently large ε so that bdu O L, and hence V L U form a concave p-end-neighborhood as above The properties for a lens-cone in non-virtually-factorable case. We may assume that each g ΓẼ is positive bi-semi-proximal by Theorem THEOREM Let O be a strongly tame convex real projective n-orbifold. Let Ẽ be an R-p-end of O S n (resp. in RP n ) with a generalized lens p-end-neighborhood. Let vẽ be the p-end vertex. Assume that π 1 (Ẽ) is non-virtually-factorable. Then ΓẼ satisfies the uniform middle eigenvalue condition with respect to vẽ, and there exists a generalized CA-lens D disjoint from vẽ with the following properties. (i) (ii) bdd D = ΛẼ is independent of the choice of D where ΛẼ is from Proposition D is strictly generalized lens-shaped. Each element g ΓẼ has an attracting fixed set in bdd intersected with the union of some great segments from vẽ in bd ΣẼ. The closure of the union of attracting fixed set is a subset of bdd A B for the top and the bottom hypersurfaces A and B. The sets are equal if ΓẼ is hyperbolic. Let l be a segment l bdo with l o Cl(U) /0 for any concave p-endneighborhood U of vẽ. Then l is in the closure in Cl(V ) of every concave or proper p-end-neighborhood V of vẽ. The set S(vẼ) of maximal segments from vẽ in Cl(V ) is independent of a concave or proper p-end neighborhood V, S(vẼ) = Cl(V ) bdo. (iii) S(g(vẼ)) = g(s(vẽ)) for g π 1 (Ẽ). (iv) Given g π 1 (O), we have ( S(g(vẼ)) ) o ( S(vẼ) ) o = /0 or else ( S(g(vẼ)) ) o ( S(vẼ) ) o = and g ΓẼ. A concave p-end neighborhood is a proper p-end neighborhood. (v) Assume that w is the p-end vertex of an R-p-end. We can choose mutually disjoint concave p-end neighborhoods for every R-p-ends. Then S o (vẽ) S(w) = /0 or S(vẼ) = S(w) (with vẽ = w)

155 PROPERLY CONVEX R-ENDS AND T-ENDS for p-end vertices vẽ and w where we defined S o (vẽ) to denote the relative interior of S(vẼ) in bdo. PROOF. We will prove these for S n first. Theorem 6.26 implies the uniform middle eigenvalue condition. (i) Let CẼ be a concave end neighborhood. Since ΓẼ acts on CẼ, CẼ is a component of the complement of a generalized lens D in a generalized R-end by definition. The action on D is cocompact and proper since we can use a foliation by great segments in a tube corresponding to Ẽ. Proposition 6.20 implies that the lens is a strict one. This implies (i). (ii) Consider any segment l in bdo with l o meeting Cl(U 1 ) for a concave p-endneighborhood U 1 of vẽ. Let T be the open tube corresponding to ΣẼ. Then O T since ΣẼ is the direction of all segments in O starting from vẽ. Let T 1 be a component of bdt 1 B containing vẽ. Then T 1 Cl(U 1 ) bdo by the definition of concave p-end neighborhoods. In the closure of U 1, an endpoint of l is in T 1. Then l o bdt since l o is tangent to T {vẽ,vẽ }. Any convex segment s from vẽ to any point of l must be in bdt. By the convexity of Cl( O), we have s Cl( O). Thus, s is in bdo since bdt Cl( O) bdo. Therefore, the segment l is contained in the union of segments in bdo from vẽ. We now suppose that l is a segment from vẽ containing a segment l 0 in Cl(U 1 ) bdo from vẽ, and we will show that l is in Cl(U 1 ) bdo. This will be sufficient to prove (ii). l o contains a point p of bdd A B, which is a subset of TẼ D. Since l Cl( O), we obtain g ΓẼ g(l) Cl( O), a properly convex subset. Hence, g ΓẼ g(l) U 1 is a distanced set, and has a distanced compact closure. Then the convex hull of the closure meets TẼ in a way contradicting Proposition 6.20 (ii) where D is ΛẼ in the proposition. Thus, l o does not meet bdd A B. Thus, l Cl(U 1 ) bd O. We define S(vẼ) as the set of maximal segments in Cl(U 1 ) bdo. We can also characterize S(vẼ) as the set of maximal segments from vẽ in bdtẽ not containing Cl(D) D in the interior. This follows since any such maximal segment have ends in Cl( O) and hence must be Cl( O). Also, S(vẼ) = Cl(U 1 ) bdo. For any other concave affine neighborhood U 2 of U 1, we have U 2 = {vẽ} D 2 D 2 {vẽ} for a generalized CA-lens D 2. Since Cl(D 2 ) D 2 equals Cl(D) D, we obtain that Cl(U 2 ) bdo = Cl(U 1 ) bdo = S(vẼ). Let U be any proper p-end-neighborhood associated with vẽ. U 1 U for a concave p- end neighborhood U 1 by Lemma Again, U 1 = {vẽ} D D {vẽ} for a generalized CA-lens D where vẽ Cl(D). Hence, Cl(U 1 ) bdo Cl(U ) bdo. Moreover, every maximal segment in S(vẼ) is in Cl(U ). We can form S (vẽ) as the set of maximal segments from vẽ in Cl(U ) bdo. Then no segment l in S (vẽ) has interior points in bdd A B as above. Thus, S(vẼ) = S (vẽ). Also, since every points of Cl(U ) bd O has a segment in the direction of bd ΣẼ, we obtain S(vẼ) = Cl(U ) bd O. (iii) By the proof above, we now characterize S(vẼ) as the set of maximal segments in bd O from vẽ ending at points of bdd A B. Since g(d) is the generalized CA-lens for

156 6.6. THE PROPERTIES OF LENS-SHAPED ENDS. 141 the the generalized lens neighborhood g(u) of g(vẽ), we obtain g(s(vẽ)) = S(g(vẼ)) for any p-end vertex vẽ. (iv) Given a concave end neighborhood CẼ of a p-end vertex vẽ, we show that for g Γ, g(cẽ) = CẼ for g ΓẼ, or else g(cẽ) CẼ = /0. This will prove the properness: First, we show that we can choose a concave end neighborhood satisfying this property. Choose a proper p-end neighborhood U of Ẽ covering an end-neighborhood of product form with compact boundary. We choose a generalized CA-lens L of a generalized lenscone so that CẼ := {vẽ} L L {vẽ} is in U by Lemma Then our choice satisfies the above property. Now, S(vẼ) o has an open neighborhood of form CẼ S(vẼ) o in O since B L is separating hypersurface. We obtain the first part of the conclusion since the intersection of the two sets implies the intersections of the neighborhoods of the sets. Now, assume that CẼ is any concave p-end neighborhood. Suppose that g(cẽ) CẼ /0 and g(cẽ) CẼ. Since CẼ is concave, each point x of bdcẽ O is contained in a sharply supporting hyperspace D so that a component C of CẼ D is in CẼ where Cl(C) v CẼ for the p-end vertex v CẼ of CẼ. Similar statements hold for g(cẽ). Since g(cẽ) CẼ /0 and g(cẽ) CẼ, it follows that bdg(cẽ) CẼ /0 or g(cẽ) bdcẽ /0. Assume the second case without the loss of generality. Let x bdc E in g(c E ) and choose D,C as above. Let Cl(C) be the closure containing vẽ of a component C of Cl( O) H for a separating hyperspace H. C bdo is a union of lines in S(vẼ). Now, H g(cẽ) contains an open neighborhood in H of x. Since H contains a point of a concave p-end neighborhood g(cẽ) of g(ẽ), it meets a points of {g(vẽ)} g(d) g(d) {g(vẽ)} and a ray from g(vẽ) in g(cẽ). We deduce that H g(cẽ) separates g(cẽ) into two components C 1 and C 2 where Cl(C 1 ) H and Cl(C 1 ) H meet ( g(s(vẽ))) o at nonempty sets. One of C 1 and C 2 is in C since C is a component of O H. Also, Cl(C) H meets the set at (Cl(C) H) bdo S(vẼ). Hence, this implies ( g(s(vẽ)) ) o S(vẼ) /0. This is a contradiction to the first part of (iv). (v) If S(vẼ) o S(w) /0, then the above argument in (iv) applies with in this situation to show that vẽ = w. [S n P] The properties of lens-cones for factorable case. Recall that a group G divides an open domain Ω if Ω/G is compact. For virtually factorable ends, we have more results. We don t require the quotient is Hausdorff. THEOREM Let O be a strongly tame properly convex real projective n-orbifold. Suppose that

157 PROPERLY CONVEX R-ENDS AND T-ENDS Cl(O) is not of form vẽ D for a totally geodesic properly convex domain D, or the holonomy group Γ is strongly irreducible. Let Ẽ be an R-p-end of the universal cover O, O S n (resp. RP n ) with a generalized lens p-end-neighborhood. Let vẽ be the p-end vertex, and ΣẼ the p-end domain of Ẽ. Suppose that the p-end holonomy group ΓẼ is virtually factorable. Then ΓẼ satisfies the uniform middle eigenvalue condition with respect to vẽ, and the following statements hold : (i) The R-p-end is totally geodesic. D i S n 1 vẽ is projectively diffeomorphic by the projection Π vẽ to totally geodesic convex domain D i in Sn ( resp. in RP n ) of dimension dimv i 1 disjoint from vẽ. ΓẼ is a cocompact subgroup of Z l0 1 l 0 i=1 Γ i where Γ i acts on D i irreducibly and trivially on D j for j i. (ii) The R-p-end is strictly lens-shaped, and each C i corresponds to a cone Ci = vẽ D i. The R-p-end has a p-end-neighborhood equal to the interior of {vẽ} D for D := Cl(D 1) Cl(D l 0 ) where the interior of D forms the boundary of the p-end neighborhood in O. (iii) The set S(vẼ) of maximal segments in bdo from vẽ in the closure of a p-endneighborhood of vẽ is independent of the p-end-neighborhood. l 0 S(vẼ) = {vẽ} Cl(D 1) Cl(D i 1) Cl(D i+1) Cl(D l 0 ). i=1 Finally, the statements (i), (ii), (iii), (iv), and (v) of Theorem 6.30 also hold. PROOF. Again the S n -version is enough. Theorem 6.26 implies the uniform middle eigenvalue condition. (i) This follows Proposition 2.49 following Benoist [17]. As in the proof of Theorem 6.30, Theorem 6.26 implies that ΓẼ satisfies the uniform middle eigenvalue condition. Proposition 6.22 implies that the CA-lens is a strict one. Theorem 6.12 implies that the distanced ΓẼ-invariant set is contained in a hyperspace P disjoint from vẽ. (i) By the uniform middle eigenvalue condition, the largest norm of the eigenvalue λ 1 (g) is strictly bigger than λẽ(g). Let U be a concave p-end-neighborhood of Ẽ in O. Let S 1,...,S l0 be the projective subspaces in general position meeting only at the p-end vertex vẽ where on the corresponding subspaces in S n 1 vẽ the factor groups Γ 1,..., Γ l0 act irreducibly by Benoist [17]. Cl( ΣẼ) S j is a properly convex domain K i by Benoist [17]. Let C i denote the union of great segments from vẽ corresponding to K i in S i for each i. The abelian center isomorphic to Z l0 1 acts as the identity on the subspace corresponding to C i in the projective space S n 1 vẽ. We denote by D i := C i P. We denote by D = D 1 D l 0 P. Also, the interior of vẽ D is a p-end neighborhood of Ẽ. This proves (i). Let U be the p-end-neighborhood of vẽ obtained in (iv). ΓẼ acts on vẽ and D 1,...,D l0. Recall that the virtual center of ΓẼ isomorphic to Z l0 1 has diagonalizable matrices acting trivially on S j for j = 1,...,l 0. For all C i, every nonidentity g in the virtual center acts as nonidentity now by the uniform middle eigenvalue condition. For each i, we can find a sequence g j in the virtual center of ΓẼ so that the premise of Proposition 2.54 are satisfied. Therefore, Cl(D i ) Cl( O) By Proposition 2.54, (ii) follows. Therefore, we obtain vẽ Cl(D 1) Cl(D i 1) Cl(D i+1) Cl(D l 0 ) = bd O Cl(U)

158 6.7. DUALITY AND LENS-SHAPED T-ENDS 143 by the middle eigenvalue conditions. (iii) follows. (ii) We need to show D o O. By Lemma 2.46, we have either D o O or D bdo. In the second case, Cl( O) = {vẽ} D since S(vẼ) bdo and D bdo. This contradicts the premise. Also, if Γ is strongly irreducible, O cannot be a strict join by Proposition Thus, this completes the proof. The final part can be shown by generalizing the proof of Theorem 6.30 to this situation. The proof statements do not change. [S n P] 6.7. Duality and lens-shaped T-ends We first discuss the duality map. We show a lens-cone p-end neighborhood of an R-pend is dual to a lens p-end neighborhood of a T-p-end. Using this we prove Theorem 6.37 dual to Theorem 6.26, i.e., Theorem Duality map. The Vinberg duality diffeomorphism induces a one-to-one correspondence between p-ends of O and O by considering the dual relationship ΓẼ and Γ Ẽ for each pair of p-ends Ẽ and Ẽ with dual p-end holonomy groups. (See Section 2.5.) Given a properly convex domain Ω in S n (resp. RP n ), we recall the augmented boundary of Ω (6.27) bd Ag Ω := {(x,h) x bdω,x h, h is an oriented sharply supporting hyperspace of Ω} S n S n. This is a closed subspace. Each x bdω has at least one sharply supporting hyperspace, an oriented hyperspace is an element of S n since it is represented as a linear functional, and an element of S n represent an oriented hyperspace in S n. We recall a duality map. (6.28) D Ω : bd Ag Ω bd Ag Ω given by sending (x,h) to (h,x) for each (x,h) bd Ag Ω. This is a diffeomorphism since D has an inverse given by switching factors. LEMMA Let Ω be the dual of a properly convex domain Ω in S n or RP n. Then (i) bdω is C 1 and strictly convex at a point p bdω if and only if bdω is C 1 and strictly convex at the unique corresponding point p. (ii) Ω is an ellipsoid if and only if so is Ω. (iii) bdω contains a properly convex domain D = P bdω open in a totally geodesic hyperspace P if and only if bdω contains a vertex p with R p (Ω) a properly convex domain. In this case, D sends the pair of p and the associated sharply supporting hyperspaces of Ω to the pairs of the totally geodesic hyperspace containing D and points of D. Moreover, D and R p (Ω) are properly convex and are projectively diffeomorphic to dual domains. PROOF. (i) bdω near p is a graph of a function f : B bdω where B is an open set in a hyperspace sharply supporting Ω at p. The C 1 -condition implies that D f : B S(R n+1 ) is well-defined. If D f is not injective in any neighborhood of p, we can deduce that there exists an identical sharply supporting hyperspace P for distinct points at bdω. P bdω is a nontrivial convex set of dimension > 0, and Ω is not strictly convex at p. Hence, D f is injective in a neighborhood of p. Now, we can apply the inverse function to obtain that bdω is C 1 also. It must be strictly convex at p since otherwise the sharply supporting

159 PROPERLY CONVEX R-ENDS AND T-ENDS A A' v P FIGURE 1. The figure for Corollary hyperspaces must be identical along a line in bdω, and the inverse map is not injective. The converse also follows by switching the role of Ω and Ω. (ii) Let R n+1 have the standard Lorentz inner product B. Let C be the open positive cone. Then the space of linear functionals positive on C is in one-to-one correspondence with vectors in C using the isomorphism C C given by φ v φ so that φ = B( v φ, ). (See [74].) (iii) Suppose that R p (Ω) is properly convex. We consider the set of hyperspaces sharply supporting Ω at p. This forms a properly convex domain as we can see the space as the projectivization of the space of linear functionals positive on C(Ω) as we explain: Let v be the vector in R n+1 in the direction of p. Then the set of linear functionals positive on C(Ω). Let V be a complementary space of v in R n+1. Let A be given as the affine subspace V+{ v} of R n+1. We choose V so that C v := C(Ω) A is a bounded convex domain in A. We give A a linear structure so that v corresponds to the origin. We denote by A this space. The set of linear functionals positive on C(Ω) and 0 at v is identical with that of linear functionals on A positive on C v : we define (6.29) C(D) := { f R n+1 f C(Ω) > 0, f ( v) = 0 } R n+1 Ĉ v := { g R n g C v > 0 }. Here indicates a linear isomorphism, which follows by the decomposition R n+1 = {t v t R} V. Define R v (C v) as the equivalence classes of properly convex segments in C v ending at v where two segments are equivalent if they agree in an open neighborhood of v. R p (Ω) is identical with R v (C v) by the projectivization S : R n+1 {O} S n. Hence R v (C v) is a properly convex open domain in S(A). Since R v (C v) is properly convex, the interior of the spherical projectivization S(Ĉ v ) S(A ) is dual to the properly convex domain R v (C v) S(A). Again we have a projection S : R n+1 {O} S n. Define D := S(C(D)) S n. Since R v (C v) corresponds to R p (Ω), and S(Ĉ v ) corresponds to D, the duality follows. Also, D bdω since points of D are oriented sharply supporting hyperspaces to Ω as we can see from (6.28). (See Proposition 6.9 also.) We will need the corollary about the duality of lens-cone and lens-neighborhoods. Recall that given a properly convex domain D in S n or RP n, the dual domain is the closure

160 6.7. DUALITY AND LENS-SHAPED T-ENDS 145 of the open set given by the collection of (oriented) hyperspaces in S n or RP n not meeting Cl(D). COROLLARY The following hold: Let L be a lens and v Cl(L) so that v L is a properly convex lens-cone. Suppose that the smooth strictly convex boundary component A of L is tangent to a segment from v at each point of bda and {v} L = {v} A. Then the dual domain of Cl(v L) is the closure of a component L 1 of L P where L is a lens and P is a hyperspace meeting L o. A corresponds to a hypersurface A bdl under the duality (6.28) and A D is the boundary of L 1 for a totally geodesic properly convex (n 1)-dimensional domain D dual to R v ({v} L). Conversely, we are given a lens L and P is a hyperspace meeting L o but not meeting the boundary of L. Let L 1 be a component of L P with smooth strictly convex boundary L 1 so that bd L 1 P. The dual of the closure of a component L 1 of L P is the closure of v L for a lens L and v L so that v L is a properly convex lens-cone. The outer boundary component A of L is tangent to a segment from v at each point of bda and v L = v A. Moreover, v Cl(A). PROOF. Let A denote the boundary component of L so that {v} L = {v} A. We will determine the dual domain ({v} L) of {v} L by finding the boundary of D using the duality map D. The set of hyperspaces sharply supporting Cl({v} L) at v forms a properly totally geodesic domain D in S n contained in a hyperspace P dual to v by Lemma Also the set of hyperspaces sharply supporting Cl({v} L) at points of A goes to the strictly convex hypersurface A in bd(v L) by Lemma 6.32 since D is a diffeomorphism. (See Remark 2.60 and Figure 1.) S := bd({v} A) A is a union of segments from v. The sharply supporting hyperspaces containing these segments go to points in D. Each point of Cl(A ) A is a limit of a sequence {p i } of points of A, corresponding to a sequence of sharply supporting hyperspheres {h i } to A. The tangency condition of A and bda implies that the limit hypersphere contains the segment in S from v. Thus, Cl(A ) A equals the set of hyperspheres containing the segments in S from v. Thus, it goes to a point of D. Thus, bda = D. We conclude (bd(v L)) = A D. Let P be the unique hyperspace containing D. Then each point of bda goes to a sharply supporting hyperspace at a point of bda distinct from P. Let L denote the dual domain of Cl(L). Since Cl(L) Cl({v} L), we obtain ({v} L) L by (2.26). Since A bdl, we obtain (bd({v} L)) A P, and A bdl. Therefore, ({v} L) is a component of L P. Moreover, A bdl 1 for a component L 1 of L P. The second item is proved similarly to the first. Then L 1 goes to a hypersurface A in the boundary of the dual domain L1 of L 1 under D. Again A is a smooth strictly convex boundary. Since bd L 1 P and L 1 is a component of L P, we have bdl 1 L 1 = Cl(L 1 ) P. This is a totally geodesic properly convex domain D. Suppose that l P be an n 2-dimensional space disjoint from L1 o. Then a space of oriented hyperspaces containing l bounding an open hemisphere containing L1 o forms a parameter dual to a convex projective geodesic in S n. An L 1 -pencil P t with ends P 0,P 1 is a parameter satisfying P t P = P 0 P,P t L1 o = /0 for all t [0,1]

161 PROPERLY CONVEX R-ENDS AND T-ENDS where P t is oriented so that it bounds a open hemisphere containing L o 1. There is a one-to-one correspondence {P P is a hyperspace that supports L 1 at points of D} {v} bda : Every supporting hyperspace P to L 1 at points of D is contained in a L 1 -parameter P t with P 0 = P,P 1 = P. v is the dual to P in S n. Each of the path P t is a geodesic segment in S n with an endpoint v. By duality map D, bdl1 is a union of A and the union of these segments. Given any hyperspace P disjoint from L1 o, we find a one-parameter family of hyperspaces containing P P. Thus, we find an L 1 -pencil family P t with P 0 = P,P 1 = P. We can extend the L 1 - pencil so that the ending hyperspace P of the L 1 -pencil meets L 1 tangentially or tangent to L 1 and P P is a supporting hyperspace of D in P. Since the hyperspaces are disjoint from L 1, the segment is in L1. Since L 1 is a properly convex domain, we can deduce that is the closure of the cone {v} A. L 1 Let L be the dual domain of L. Since L L 1, we obtain L L1 by (2.26). Since L 1 bdl, we obtain A bdl by the duality map D. We obtain that L A Cl({v} A). Let B be the image of the other boundary component B of L under D. We take a sharply supporting hyperspace P y at y B. Then P y P is disjoint from Cl(D) by convexity. Then we find an L 1 -pencil P t of hyperspaces containing P y P with P 0 = P y,p 1 = P. This L 1 -pencil goes into the segment from v to a point of B under the duality. We can extend the L 1 -pencil so that the ending hyperspace meets L 1 tangentially. The dual pencil is a segment from v to a point of A. Thus, each segment from v to a point of A meets B. Thus, L o A B is a lens of the lens cone {v} A. This completes the proof The duality of T-ends and properly convex R-ends. Let Ω be the properly convex domain covering O. For a T-end E, the totally geodesic ideal boundary S E of E is covered by a properly convex open domain in bdω corresponding to a T-p-end Ẽ. We denote it by SẼ. PROPOSITION Let O be a strongly tame properly convex real projective orbifold with R-ends or T-ends. Then the dual real projective orbifold O is also strongly tame and has the same number of ends so that there exists a one-to-one correspondence C between the set of ends of O and the set of ends of O. C restricts to such a one between the subset of horospherical ends of O and the subset of horospherical ones of O. C restricts to such a one between the set of T-ends of O with the set of ends of properly convex R-ends of O. The ideal boundary component SẼ for a T-p-end Ẽ is projectively diffeomorphic to the properly convex open domain dual to the domain ΣẼ for the corresponding R-p-end Ẽ of Ẽ. C restricts to such a one between the subset of all properly convex R-ends of O and the subset of all T-ends of O. Also, ΣẼ of an R-p-end is projectively dual to the ideal boundary component SẼ for the corresponding dual T-p-end Ẽ of Ẽ. PROOF. We prove for the S n -version. By the Vinberg duality diffeomorphism of Theorem 2.61, O is also strongly tame. Let O be the universal cover of O. Let O be the dual domain. The first item follows by the fact that this diffeomorphism sends pseudo-ends neighborhoods to pseudo-end neighborhoods.

162 6.7. DUALITY AND LENS-SHAPED T-ENDS 147 Let Ẽ be a horospherical R-p-end with x as the end vertex. Since there is a subgroup of a cusp group acting on Cl( O) with a unique fixed point, the intersection of the unique sharply supporting hyperspace h with Cl( O) at x is a singleton {x}. (See Theorem 4.7.) The dual subgroup is also a cusp group and acts on Cl( O ) with h fixed. So the corresponding O has the dual hyperspace x of x as the unique intersection at h dual to h at Cl( O ). Hence x is the vertex of a horospherical end. An R-p-end Ẽ of O has a p-end vertex vẽ. ΣẼ is a properly convex domain in S n 1 vẽ. The space of sharply supporting hyperspaces of O at vẽ forms a properly convex domain of dimension n 1 since they correspond to hyperspaces in S n 1 vẽ not intersecting ΣẼ. Under the duality map DO in Proposition 2.58, (v Ẽ,h) for a sharply supporting hyperspace h is sent to (h,v Ẽ ). Lemma 6.32 shows that h is a point in a properly convex n 1- dimensional domain bdo P for P = v Ẽ, a hyperspace. The map D Ω sends points of bd Ag Ω to bd Ag Ω. Thus, SẼ bdω, and Ẽ is a totally geodesic end with SẼ dual to ΣẼ. This proves the fourth item. The third item follows similarly. [SS n ] REMARK We also remark that the map induced on the set of pseudo-ends of O to that of O by DO is compatible with the Vinberg diffeomorphism. This easily follows by Proposition 6.7 of [74] and the fact that the level set S x R n+1 of the Koszul-Vinberg function is asymptotic to the boundary of O. Thus, the hyperspace in R n+1 corresponding to the sharply supporting hyperspace of a p-end vertex is approximated by a tangent hyperspace to S x in R n+1. DO sends a point p of S x to the linear form corresponding to the tangent hyperspace of S x at p. (See Chapter 6 of Goldman [74].) Thus, points in the R-p-end neighborhoods go to points of T-p-end neighborhoods and vice versa. C restricts to a correspondence between the lens-shaped R-ends with lens-shaped T- ends. See Corollary 6.38 for detail. PROPOSITION Let O be a strongly tame properly convex real projective orbifold. The following conditions are equivalent : (i) A properly convex R-end of O satisfies the uniform middle-eigenvalue condition. (ii) The corresponding totally geodesic end of O satisfies this condition. PROOF. The items (i) and (ii) are equivalent by considering (6.1) and (6.2). We now prove the dual to Theorem For this we do not need the triangle condition or the reducibility of the end. THEOREM Let O be a strongly tame properly convex real projective orbifold. Let SẼ be a totally geodesic ideal boundary component of a T-p-end Ẽ of O. Then the following conditions are equivalent : (i) The end holonomy group of Ẽ satisfies the uniform middle-eigenvalue condition with respect to the T-p-end structure of Ẽ. (ii) SẼ has a CA-lens neighborhood in an ambient open manifold containing O with cocompact action of π 1 (Ẽ), and hence Ẽ has a lens-shaped p-end-neighborhood in O. PROOF. It suffices to prove for S n. Assuming (i), the existence of a lens neighborhood follows from Theorem Assuming (ii), we obtain a totally geodesic (n 1)-dimensional properly convex domain SẼ in a subspace S n 1 where ΓẼ acts on. Let U be a CA-lens-neighborhood of it where ΓẼ acts on. Then since U is a neighborhood, the sharply supporting hemisphere at

163 PROPERLY CONVEX R-ENDS AND T-ENDS each point of Cl(SẼ) SẼ is now transversal to S n 1. Let P be the hyperspace containing SẼ, and let U 1 be the component of U P. Then the dual U1 is a lens-cone by the second part of Corollary The dual U of U is a CA-lens contained in a lens-cone U1 where Γ E acts on U. We apply the part (i) (ii) of Theorem Theorems 6.26 and 6.37 and Propositions 6.34 and 6.36 imply COROLLARY Let O be a strongly tame properly convex real projective orbifold and let O be its dual orbifold. The dual end correspondence C restricts to a correspondence between the generalized lens-shaped R-ends with lens-shaped T-ends and horospherical ends to themselves. If O satisfies the triangle condition or every end is virtually factorable, C restricts to a correspondence between the lens-shaped R-ends with lens-shaped T-ends and horospherical ends to themselves.

164 CHAPTER 7 Application: The openness of the lens properties, and expansion and shrinking of end neighborhoods This chapter lists applications of the main theory of Part 2, which are results we need in Part 3. Let O be a strongly tame properly convex real projective orbifold with generalized lens-shaped or horospherical R-ends or lens-shaped T-end and satisfy (IE) and (NA). In Section 7.1, we show that the lens-shaped property is a stable property under the change of holonomy representations. In Section 7.1.1, we prove a minor extension of Koszul s openness for bounded manifolds, well-known to many people. In Section 7.2, we will define limits sets of ends and discuss the properties. We obtain the exhaustion of O by a sequence of p-end-neighborhoods of O, and some other results. We go to Section 7.3. We prove the strong irreducibility of O; that is, Theorem The openness of lens properties. As conditions on representations of π 1 (Ẽ), the condition for generalized lens-shaped ends and one for lens-shaped ends are the same. Given a holonomy group of π 1 (Ẽ) acting on a generalized lens-shaped cone p-end neighborhood, the holonomy group satisfies the uniform middle eigenvalue condition by Theorem We can find a lens-cone by choosing our orbifold to be T o vẽ /π 1 (Ẽ) by Proposition Let Hom E (π 1 (Ẽ),SL ± (n + 1,R)) (resp. Hom E (π 1 (Ẽ),PGL(n + 1,R))) denote the space of representations of the fundamental group of an (n 1)-orbifold ΣẼ. Recall Definition 6.21 for strictly generalized lens-shaped R-ends. A (resp. generalized) lens-shaped representation for an R-end fundamental group is a representation acting on a (resp. generalized) lens-cone. THEOREM 7.1. Let O be a strongly tame properly convex real projective orbifold. Assume that the universal cover O is a subset of S n (resp. RP n ). Let Ẽ be a properly convex R-p-end of the universal cover O. Then (i) Ẽ is a generalized lens-shaped R-end if and only if Ẽ is a strictly generalized lens-shaped R-end. (ii) The subspace of generalized lens-shaped representations of an R-end is open in Hom E (π 1 (Ẽ),SL ± (n + 1,R)) (resp. Hom E (π 1 (Ẽ),PGL(n + 1,R))). Finally, if O is properly convex and satisfies the triangle condition or every end is virtually factorable, then we can replace the term generalized lens-shaped to lens-shaped in each of the above statements. PROOF. We will assume O S n first. (i) If π 1 (Ẽ) is non-virtually-factorable, then the equivalence is given in Theorem 6.30 (i), and if π 1 (Ẽ) is virtually factorable, then it is in Theorem 6.31 (iv). (ii) Let µ be a representation π 1 (Ẽ) SL ± (n + 1,R) associated with a generalized lens-cone. By Theorem 6.6, we obtain a CA-lens K in T vẽ with smooth convex boundary 149

165 APPLICATIONS components A B since T vẽ itself satisfies the triangle condition although it is not properly convex. (Note we don t need K to be in O for the proof.) K/µ(π 1 (Ẽ)) is a compact orbifold whose boundary is the union of two closed n- orbifold components A/µ(π 1 (Ẽ)) B/µ(π 1 (Ẽ)). Suppose that µ is sufficiently near µ. We may assume that vẽ is fixed by conjugating µ by a bounded projective transformation. By considering the radial segments in K, we obtain a foliation by radial lines in K also. By Proposition 7.4, applying Proposition 7.5 to the both boundary components of the CA-lens, we obtain a lens-cone in a tube domain T vẽ in general different from the original one. This implies that the sufficiently small change of holonomy keep Ẽ to have a concave p-end neighborhood. This completes the proof of (ii). The final statement follows by Lemma [S n T] A lens-shaped representation for a T-end fundamental group is a representation acting on a lens containing the image of SẼ satisfying the lens condition with respect to that lens. THEOREM 7.2. Let O be a strongly tame properly convex real projective orbifold. Assume that the universal cover O is a subset of S n (resp. of RP n ). Let Ẽ be a T-p-end of the universal cover O. Let Hom E (π 1 (Ẽ),SL ± (n+1,r)) (resp. Hom E (π 1 (Ẽ),PGL(n+1,R))) be the space of representations of the fundamental group of an n 1-orbifold ΣẼ. Then the subspace of lens-shaped representations of a T-p-end is open. PROOF. By Theorem 6.37, the condition of the lens T-p-end is equivalent to the uniform middle eigenvalue condition for the end. Proposition 6.36 and Theorems 6.7 and 7.1 complete the proof. [S n T] COROLLARY 7.3. We are given a properly convex end Ẽ of a strongly tame convex orbifold O. Assume that O S n (resp. O RP n ). Then the subset of Hom E (π 1 (Ẽ),SL ± (n + 1,R)) (resp.hom E (π 1 (Ẽ),PGL(n + 1,R))) consisting of representations satisfying the uniform middle-eigenvalue condition with respect to some choices of fixed points or fixed hyperplanes of the holonomy group is open. PROOF. For R-p-ends, this follows by Theorems 6.26 and 7.1. For T-p-ends, this follows by dual results: Theorem 6.37 and Theorems 7.2. [S n T] An extension of Koszul s openness. Here, we state and prove a well-known minor modification of Koszul s openness result. A radial affine connection is an affine connection on R n+1 {O} invariant under a scalar dilatation S t : v t v for every t > 0. PROPOSITION 7.4 (Koszul). Let M be a properly convex real projective compact n- orbifold with strictly convex boundary. Let h : π 1 (M) PGL(n + 1,R) (resp. SL ± (n + 1,R)) denote the holonomy homomorphism acting on a properly convex domain Ω h in RP n (resp. in S n ). Assume that M is projectively diffeomorphic to Ω h /h(π 1 (M)). Then there exists a neighborhood U of h in Hom(π 1 (M),PGL(n+1,R)) (resp. Hom(π 1 (M),SL ± (n+ 1,R))) so that every h U acts on a properly convex domain Ω h so that Ω h /h (π 1 (M)) is a compact properly convex real projective n-orbifold Ω h /h (π 1 (M)) with strictly convex boundary. Also, Ω h /h (π 1 (M)) is diffeomorphic to M.

166 7.1. THE OPENNESS OF LENS PROPERTIES. 151 PROOF. We prove for S n. Let Ω h be a properly convex domain covering M. We may modify M by pushing M inward: Let Ω h be the inverse image of M in M. Then M and Ω h are properly convex by Lemma The linear cone C(Ω o h ) Rn+1 = Π 1 (Ω o h ) over Ωo h has a smooth strictly convex Hessian function V by Vey [134] or Vinberg [91]. Let C(Ω h ) denote the linear cone over Ω h. We extend the group µ(π 1 (M)) by adding a transformation γ : v 2 v to C(Ω o h ). For the fundamental domain F of C(Ω h ) under this group, the Hessian matrix of V restricted to F C(Ω h ) has a lower bound. Also, the boundary C(Ω h ) is strictly convex in any affine coordinates in any transversal subspace to the radial directions at any point. Let N be a compact orbifold C(Ω h )/ µ(π 1(Ẽ)),γ with a flat affine structure. Note that S t, t R +, becomes an action of a circle on M. The change of representation h to n : π 1 (M) SL ± (n + 1,R) is realized by a change of holonomy representations of M and hence by a change of affine connections on C(Ω h ). Since S t commutes with the images of h and h, S t still gives us a circle action on N with a different affine connection. We may assume without loss of generality that the circle action is fixed and N is invariant under this action. Thus, N is a union of B 1,...,B m0 that are n-ball times circles foliated by circles that are flow arcs of S t. We can change the affine structure on N to a one with the holonomy group h (π 1 (Ẽ)),γ by by local regluing of balls B 1,...,B m0 as in [43]. We assume that S t still gives us a circle affine action since γ is not changed. We may assume that N and N are foliated by circles that are flow curves of the circle action. The change corresponds to a sufficiently small C r -change in the affine connection for r 2 as we can see from [43]. Now, the strict positivity of the Hessian of V in the fundamental domain, and the boundary convexity are preserved. Let C(Ω h ) denote the universal cover of N with the new affine connection. Thus, C(Ω h ) is also a properly convex affine cone by Koszul s work [100]. Also, it is a cone over a properly convex domain Ω h in Sn. [S n T] We denote by PGL(n+1,R) v the subgroup of PGL(n+1,R) fixing a point v,v RP n. and denote by SL ± (n + 1,R) v the subgroup of SL ± (n + 1,R) fixing a point v, v S n. PROPOSITION 7.5. Let T be a tube domain over a properly convex domain Ω RP n ( resp. S n 1 ). Let B be a strictly convex hypersurface bounding a properly convex domain in a tube domain T. Let v be a vertex of T. B meets each radial ray in T from v transversally. Assume that a projective group Γ acts on Ω properly discontinuously and cocompactly. Then there exists a neighborhood of I in Hom(Γ,PGL(n + 1,R) v ) (resp. Hom(π 1 (M),SL ± (n + 1,R) v )) where every element h acts on a strictly convex hypersurface B h in a tube domain T h meeting each radial ray at a unique point and bounding a properly convex domain in T h. PROOF. We assume first that B,T S n. For sufficiently small neighborhood V of h in Hom(Γ,SL ± (n + 1,R) v ), h(γ), h V acts on a properly convex domain Ω h properly discontinuously and cocompactly by Koszul [100]. Let T h denote the tube over Ω h. Since B/Γ is a compact orbifold, we choose V V so that for the projective connections on a compact neighborhood of B/Γ corresponding to elements of V, B/Γ is still strictly convex and transversal to radial lines. For each h V, we obtain an immersion to a strictly convex domain ι h : B T h transversal to radial lines. Let p Th : T h Ω h denote the projection with fibers equal to the radial lines. Since p Th ι h is proper immersion to Ω h, the result follows. [S n T]

167 APPLICATIONS DEFINITION The end and the limit sets. Define the limit set Λ(Ẽ) of an R-p-end Ẽ with a generalized p-end-neighborhood to be bdd D for a generalized CA-lens D of Ẽ in S n (resp. RP n ). This is identical with the set ΛẼ in Theorem The limit set Λ(Ẽ) of a lens-shaped T-p-end Ẽ to be Cl( SẼ) SẼ for the ideal boundary component SẼ of Ẽ. The limit set of a horospherical end is the set of the end vertex. The definition does depend on whether we work on S n or RP n. However, by Proposition 2.40, there are always straightforward one-to-one correspondences. We remark that this may not equal to the closure of the union of the attracting fixed set for factorable cases. COROLLARY 7.7. Let O be a strongly tame convex real projective n-orbifold where O S n (resp. RP n ). Let U be a p-end-neighborhood of Ẽ where Ẽ is a lens-shaped T-pend or a generalized lens-shaped or lens-shaped or horospherical R-p-end. Then Cl(U) bdo equals Cl( SẼ) or Cl( S(vẼ)) or {vẽ} depending on whether Ẽ is a lens-shaped T-pend or a generalized lens-shaped or lens-shaped or horospherical R-p-end. Furthermore, this set is independent of the choice of U and so is the limit set Λ(Ẽ) of Ẽ. PROOF. We first assume O S n. Let Ẽ be a generalized lens-shaped R-p-end. Then by Theorem 6.26, Ẽ satisfies the uniform middle eigenvalue condition. Suppose that π 1 (Ẽ) is not virtually factorable. Let L b denote bdt vẽ L for a distanced compact convex set L where ΓẼ acts on. L b = ΛẼ by Proposition Since S(vẼ) is an h(π 1 (Ẽ))-invariant set and the convex hull of bd S(vẼ) is a distanced compact convex set by the proper convexity of ΣẼ. Theorem 6.30 shows that the limit set is determined by the set L b in S(vẼ), and Cl(U) bdo = S(vẼ). By Proposition 6.20, each point of L b is a limit of some g i (x) for x D for a generalized CA-lens D. Suppose now that π 1 (Ẽ) is factorable. Then by Theorem 6.31, Ẽ is a totally geodesic R-p-end. Proposition 6.20 again implies the result. Let Ẽ be a T-p-end. Theorems 6.37 and 5.21 imply Cl(A) A Cl( SẼ) for A = bdl O for a CA-lens neighborhood L by the strictness of the lens. Thus, Cl(U) bd O equals Cl( SẼ). For horospherical, we simply use the definition to obtain the result. [S n T] Since a properly convex domain is in a bounded subset of an affine subspace, the limit set defined for S n coincide with that defined for RP n under the double covering map. Hence, we regard these as being well-defined for the S n or RP n cases. DEFINITION 7.8. An SPC-structure or a stable properly-convex real projective structure on an n-orbifold is a real projective structure so that the orbifold with stable irreducible holonomy. That is, it is projectively diffeomorphic to a quotient orbifold of a properly convex domain in S n (resp. in RP n ) by a discrete group of projective automorphisms that is stable and irreducible. DEFINITION 7.9. Suppose that O has an SPC-structure. Let Ũ be the inverse image in O in S n (resp. in RP n ) of the union U of some choice of a collection of disjoint end

168 7.2. THE END AND THE LIMIT SETS. 153 neighborhoods of O. If every straight arc in the boundary of the domain O and every non- C 1 -point is contained in the closure of a component of Ũ for some choice of U, then O is said to be strictly convex with respect to the collection of the ends. And O is also said to have a strict SPC-structure with respect to the collection of ends. Corollary 7.7 show the independence of the definition with respect to the choice of the end-neighborhoods. COROLLARY Suppose that O is a strongly tame strictly SPC-orbifold with generalized lens-shaped R-ends or lens-shaped T-ends or horospherical ends. Let O is a properly convex domain in RP n ( resp. in S n ) covering O. Choose any disjoint collection of end neighborhoods in O. Let U denote their union. Let p O : O O denote the universal cover. Then any segment or a non-c 1 -point of bdo is contained in the closure of a component of p 1 O (U) for any choice of U. PROOF. We first assume O S n. By the definition of a strict SPC-orbifold, any segment or a non-c 1 -point has to be in the closure of a p-end neighborhood. Corollary 7.7 proves the claim. [S n P] Convex hulls of ends. We will sharpen Corollary 7.7 and the convex hull part in Lemma One can associate a convex hull C H (Ẽ) of a p-end Ẽ of O as follows: For horospherical p-ends, the convex hull of each is defined to be the set of the end vertex actually. The convex hull of a lens-shaped totally geodesic p-end Ẽ is the closure Cl( SẼ) the totally geodesic ideal boundary component SẼ corresponding to Ẽ. For a generalized lens-shaped p-end Ẽ, the convex hull of Ẽ is the convex hull of S(vẼ) in Cl( O). The first two equal Cl(U) bdo for any p-end neighborhood U of Ẽ by Corollary 7.7. Corollary 7.7 and Proposition 7.11 imply that the convex hull of an end is well-defined. For a lens-shaped p-end Ẽ with a p-end vertex vẽ, the convex hull of the end Ẽ is defined as I(Ẽ) := C H ( S(vẼ)). We can also characterize as the intersection I(Ẽ) = C H (Cl(U 1 )) U 1 U for the collection U of p-end neighborhoods U 1 of vẽ by Proposition We define S I(Ẽ) as the set of endpoints of maximal rays from vẽ ending at I(Ẽ) and in direction of ΣẼ. PROPOSITION Let O be a strongly tame properly convex real projective orbifold with radial ends or lens-shaped totally geodesic ends and satisfy (IE) and (NA). Let Ẽ be a generalized lens-shaped R-p-end and vẽ an associated p-end vertex. Let I(Ẽ) be the convex hull of Ẽ. (i) Suppose that Ẽ is a lens-shaped radial p-end. Then S I(Ẽ) = I(Ẽ) O, and S I(Ẽ) is contained in the union of a CA-lens in a lens-shaped p-end neighborhood.

169 APPLICATIONS (ii) I(Ẽ) contains any concave p-end-neighborhood of Ẽ and I(Ẽ) = C H (Cl(U)) I(Ẽ) O = C H (Cl(U)) O I(Ẽ) = S I(Ẽ) S(vẼ), with S I(Ẽ) S(vẼ) = /0, for a concave p-end neighborhood U of Ẽ. Thus, I(Ẽ) has a nonempty interior. (iii) Each segment from vẽ maximal in Cl( O) meets the set S I(Ẽ) exactly once and S I(Ẽ)/ΓẼ is homeomorphic to Σ E. If the end is lens-shaped, it is isotopic to Σ E. (iv) There exists a nonempty interior of the convex hull I(Ẽ) of Ẽ where ΓẼ acts so that I(Ẽ) o /ΓẼ is diffeomorphic to the end orbifold times an interval. PROOF. Assume first that O S n. (i) Suppose that Ẽ is lens-shaped. We define S 1 as the set of 1-simplices with endpoints in segments in S(vẼ) and we inductively define S i to be the set of i-simplices with faces in S i 1. Then I(Ẽ) = Notice that S I(Ẽ) is the union σ S 1 S 2 S m σ. σ S 1 S 2 S m,σ o S I(Ẽ) since each point of S I(Ẽ) is contained in the interior of a simplex which lies in S I(Ẽ) by the convexity of I(Ẽ). If σ S 1 with σ S I(Ẽ), then its endpoint must be in an endpoint of a segment in S(vẼ): otherwise, σ o is in the interior of I(Ẽ). If an interior point of σ is in a segment in S(vẼ), then the vertices of σ are in S(vẼ) by the convexity of Cl(R vẽ ( O)). Hence, if σ o bdi(ẽ) meets O, then σ o is contained in a CA-lens-shaped domain L as the vertices of σ is in bdl L by the convexity of L. Now by induction on S i, i > 1, we can verify (i) since any simplex with boundary in the union of subsimplices in the CA-lens-domain is in the CA-lens-domain by convexity. Also, each point of S I(Ẽ) is in L o Σ. Hence S I(Ẽ) bdi(ẽ) O. Conversely, a point of I(Ẽ) O is an end point of Hence, S I(Ẽ) = I(Ẽ) O. (ii) Since I(Ẽ) contains the segments in S(vẼ) and is convex, and so does a concave p- end neighborhood U, we obtain bdu I(Ẽ): Otherwise, let x be a point of bdu bdi(ẽ) O where some neighborhood in bdu is not in I(Ẽ). Then since bdu is a union of a convex hypersurface bdu O and S(vẼ), each sharply supporting hyperspace at x of the convex set bdu O meets a segment in S(vẼ) in its interior. This is a contradiction since x must be then in I(Ẽ) o. Thus, U I(Ẽ). Thus, C H (Cl(U)) I(Ẽ). Conversely, since Cl(U) S(vẼ) by Theorems 6.30 and 6.31, we obtain that C H (Cl(U)) I(Ẽ). Since I(Ẽ) is a convex domain foliated by maximal segments from vẽ, the last equation follows. (iii) Each point of it meets a maximal segment from vẽ in the end but not in S(vẼ) at exactly one point since a maximal segment must leave the lens cone eventually. Thus S I(Ẽ) is homeomorphic to an (n 1)-cell and the result follows. (iv) This follows from (iii) since we can use rays from x meeting bdi(ẽ) O at unique points and use them as leaves of a fibration. [S n P] σ o

170 7.2. THE END AND THE LIMIT SETS. 155 Bd I I S(x) FIGURE 1. The structure of a lens-shaped p-end. µ Expansion of lens or horospherical p-end-neighborhoods. LEMMA Let O have a strongly tame properly convex real projective structure Let U 1 be a p-end neighborhood of a horospherical or a lens-shaped R-p-end Ẽ with the p-end vertex vẽ; or Let U 1 be a lens-shaped p-end neighborhood of a T-p-end Ẽ. Let ΓẼ denote the p-end holonomy group corresponding to Ẽ. Then we can construct a sequence of lens-cone or lens p-end neighborhoods U i, i = 1,2,..., where U i U j O for i < j where the following hold : Given a compact subset of O, there exists an integer i 0 such that U i for i > i 0 contains it. The Hausdorff distance between U i and O can be made as small as possible, i.e., ε > 0, J,J > 0, so that d H (U i, O) < ε for i > J. There exists a sequence of convex open p-end neighborhoods U i of Ẽ in O so that (U i U j )/ΓẼ for a fixed j and i > j is diffeomorphic to a product of an open interval with the end orbifold. We can choose U i so that bdu i O is smoothly embedded and strictly convex with Cl(bdU i ) O Λ(Ẽ). PROOF. Suppose that O S n first. Suppose that Ẽ is a lens-shaped R-end first. Let U 1 be a lens-cone. Take a union of finitely many geodesic leaves L from vẽ in O of do - length t outside the lens-cone U 1 and take the convex hull of U 1 and ΓẼ(L) in O. Denote the result by Ω t. Thus, the endpoints of L not equal to vẽ are in O. We claim that bdω t O is a connected (n 1)-cell, bdω t O/ΓẼ is a compact (n 1)-orbifold diffeomorphic to ΣẼ, and

171 APPLICATIONS bdu 1 O bounds a compact orbifold diffeomorphic to the product of a closed interval with (bdω t O)/ΓẼ: First, each leaf of g(l),g ΓẼ for l in L is so that any converging subsequence of {g i (l)},g i ΓẼ, converges to a segment in S(vẼ) for a sequence {g i } of mutually distinct elements. This follows since a limit is a segment in bdo with an endpoint vẽ and must belong to S(vẼ) by Theorems 6.30 and Let S 1 be the set of segments with endpoints in ΓẼ(L) S(vẼ). We define inductively S i to be the set of simplices with sides in S i 1. Then the convex hull of ΓẼ(L) in Cl( O) is a union of S 1 S m. We claim that for each maximal segment s in Cl( O) from vẽ not in S(vẼ), s o meets bdω t O at a unique point: Suppose not. Then let v be its other endpoint of s in bdo with s o bdω t O = /0. Thus, v bdω t. Now, v is contained in the interior of a simplex σ in S i for some i. Since σ o bdo /0, σ bdo by Lemma Since the endpoints ΓẼ(L) are in O, the only possibility is that the vertices of σ are in S(vẼ). Also, σ o is transversal to radial rays since otherwise v is not in bdo. Thus, σ o projects to an open simplex of same dimension in ΣẼ. Since U 1 is convex and contains S(vẼ) in its boundary, σ is in the lens-cone Cl(U 1 ). Since a lens-cone has boundary a union of a strictly convex open hypersurface A and S(vẼ), and σ o cannot meet A tangentially, it follows that σ o is in the interior of the lens-cone. and no interior point of σ is in bdo, a contradiction. Therefore, each maximal segment s from vẽ meets the boundary bdω t O exactly once. As in Lemma 6.24, bdω t O contains no line segment ending in bdo. The strictness of convexity of bdω t O follows by smoothing as in the proof of Proposition By taking sufficiently many leaves for L with do -lengths t sufficiently large, we can show that any compact subset is inside Ω t. From this, the final item follows. The first three items now follow if Ẽ is an R-end. Suppose now that Ẽ is horospherical and U 1 is a horospherical p-end neighborhood. We can smooth the boundary to be strictly convex. Call the set Ω t where t is a parameter measuring the distance from U 1. ΓẼ is in a parabolic or cusp subgroup of a conjugate of SO(n,1) by Theorem 4.8. By taking L sufficiently densely, we can choose similarly to above a sequence Ω i of strictly convex horospherical open sets at vẽ so that eventually any compact subset of O is in it for sufficiently large i. Suppose now that Ẽ is totally geodesic. Now we use the dual domain O and the group Γ Ẽ. Let v Ẽ denote the vertex dual to the hyperspace containing SẼ. By the diffeomorphism induced by great segments with endpoints v, we obtain Ẽ (bd O S(vẼ ))/Γ Ẽ = ΣẼ /Γ Ẽ, a compact orbifold. Then we obtain U i containing O in TẼ by taking finitely many hypersphere F i disjoint from O but meeting TẼ. Let H i be the open hemisphere containing O bounded by F i. Then we form U 1 := g ΓẼ g(h i ). By taking more hyperspheres, we obtain a sequence U 1 U 2 U i U i+1 O so that Cl(U i+1 ) U i and i Cl(U i ) = Cl( O ).

172 7.2. THE END AND THE LIMIT SETS. 157 That is for sufficiently large hyperspaces, we can make U i disjoint from any compact subset disjoint from Cl( O ). Now taking the dual U of U i and by equation (2.26) we obtain i U 1 U 2 U i U i+1 O. Then Ui O is an increasing sequence eventually containing all compact subset of O. This completes the proof for the first three items. The fourth item follows from Corollary 7.7. [S n P] Shrinking of lens and horospherical p-end-neighborhoods. We now discuss the shrinking of p-end-neighborhoods. These repeat some results. COROLLARY Suppose that O is a strongly tame properly convex real projective orbifold and let O be a properly convex domain in S n (resp. RP n ) covering O. Then the following statements hold : (i) If Ẽ is a horospherical R-p-end, every p-end-neighborhood of Ẽ contains a horospherical p-end-neighborhood. (ii) Suppose that Ẽ is a generalized lens-shaped or lens-shaped R-p-end. Let I(Ẽ) be the convex hull of S(vẼ), and let V be a p-end-neighborhood V where (bdv O)/π 1 (Ẽ) is a compact orbifold. If V o I(Ẽ) O, then V contains a lens-cone p-end neighborhood of Ẽ, and a lens-cone contains O properly. (iii) If Ẽ is a generalized lens-shaped R-p-end or satisfies the uniform middle eigenvalue condition, every p-end-neighborhood of Ẽ contains a concave p-end-neighborhood. (iv) Suppose that Ẽ is a lens-shaped T-p-end or satisfies the uniform middle eigenvalue condition. Then every p-end-neighborhood contains a lens p-end-neighborhood L with strictly convex boundary in O. (iv) We can choose a collection of mutually disjoint end neighborhoods for all ends that are lens-shaped or concave end-neighborhood or a hospherical ones. PROOF. Suppose that O S n first. (i) Let vẽ denote the R-p-end vertex corresponding to Ẽ. By Theorem 4.8, we obtain a conjugate G of a subgroup of a parabolic or cusp subgroup of SO(n,1) as the finite index subgroup of h(π 1 (Ẽ)) acting on U, a p-end-neighborhood of Ẽ. We can choose a G-invariant ellipsoid of d-diameter ε for any ε > 0 in U containing vẽ. (ii) This follows from Proposition 6.22 since the convex hull of S(vẼ) contains a generalized lens with the right properties. Let U be a concave p-end neighborhood of Ẽ. We realize O to be inside a tubular domain T corresponding to ΣẼ. We take a fundamental domain F in ΣẼ and take a tube T F over F with vertex vẽ. The set O T F U is a compact set to be denoted J. We take the convex hull of g ΓẼ g(j). By Lemma 6.17 and 6.19, we obtain that g ΓẼ g(j) has accumulation points only in Λ(Ẽ). Hence, we can take a convex hull L of the set. Then vẽ L gives us a generalized lens. (iii) This was proved in Proposition (iv) The existence of a lens-shaped p-end neighborhood of SẼ follows from Theorem (v) We choose a mutually disjoint end neighborhoods for all ends. Then we choose the desired ones by the above. [S n T] The mc-p-end neighborhoods. The mc-p-end neighborhood will be useful in other papers.

173 APPLICATIONS DEFINITION Let Ẽ be a lens-shaped R-end. Let C H (Λ(Ẽ)) denote the convex hull of Λ(Ẽ). Let U be any p-end neighborhood U of Ẽ containing C H (Λ(Ẽ)) O. We define a maximal concave p-end neighborhood or mc-p-end-neighborhood U to be one of the components of U C H (Λ(Ẽ)) containing a p-end neighborhood of Ẽ. The closed maximal concave p-end neighborhood is Cl(U) O. An ε-do -neighborhood U of a maximal concave p-end neighborhood is called an ε-mc-p-end-neighborhood.. In fact, these are independent of choices of U. Note that a maximal concave p-end neighborhood U is uniquely determined since so is Λ(Ẽ). Each radial segment s in O from vẽ meets bdu O at a unique point since the point s bdu is in an n 1)-dimensional ball D = P U for a hyperspace P sharply supporting C H (Λ(Ẽ)) with Cl(D) S(vẼ). LEMMA Let D be an i-dimensional totally geodesic compact convex domain, i 1. Let Ẽ be a generalized lens-shaped R-p-end with the p-end vertex vẽ. Suppose D S(vẼ). Then D V for a maximal concave p-end neighborhood V, and for sufficiently small ε > 0, an ε-do -neighborhood of Do is contained in V for any ε-mc-p-end neighborhood V. PROOF. Suppose that O S n first. Assume that U is a generalized lens-cone of vẽ. Then Λ(Ẽ) is the set of endpoints of segments in S(vẼ) which are not vẽ by Theorems 6.30 and Let P be the subspace spanned by D {vẽ}. Since D,Λ(Ẽ) P S(vẼ) P, and D P is closer than Λ(Ẽ) P from vẽ, it follows that P Cl(U) D has a component C 1 containing vẽ and Cl(P Cl(U) C 1 ) contains Λ(Ẽ) P. Hence Cl(P Cl(U) C 1 ) C H (Λ(Ẽ)) P by the convexity of Cl(P Cl(U) C 1 ). Since C H (Λ(Ẽ)) P is a convex set in P, we have only two possibilities D is disjoint from C H (Λ(Ẽ)) o or D contains C H (Λ(Ẽ)) P. Let V be an mc-p-end neighborhood of U. Since Cl(V ) contains the closure of the component of U C H (Λ(Ẽ)) whose closure contains vẽ, it follows that Cl(V ) contains D. Since D is in Cl(V ), the boundary bdv O of the ε-mc-p-end neighborhood V do not meet D. Hence D o V. [S n P] The following gives us a characterization of ε-mc-p-end neighborhood of Ẽ. COROLLARY Let O be a properly convex real projective orbifold with generalized lens-shaped or horospherical ends and satisfies (IE) and (NA). Let Ẽ be a generalized lens-shaped R-end. Then the following statements hold: (i) A concave p-end neighborhood of Ẽ is always a subset of an mc-p-end-neighborhood of the same R-p-end. (ii) The closed mc-p-end-neighborhood of Ẽ is the closure in O of a union of all concave end neighborhoods of Ẽ. (iii) The mc-p-end-neighborhood V of Ẽ is a proper p-end neighborhood, and covers an end-neighborhood with compact boundary in O. (iv) An ε-mc-p-end-neighborhood of Ẽ for sufficiently small ε > 0 is a proper p-end neighborhood.

174 7.2. THE END AND THE LIMIT SETS. 159 (v) For sufficiently small ε > 0, the image end-neighborhoods in O of ε-mc-p-end neighborhoods of R-p-ends are mutually disjoint or identical. PROOF. Suppose first that O S n. (i) Since the limit set Λ(Ẽ) is in any generalized CA-lens L by Corollary 7.7, a generalized lens-cone p-end neighborhood U of Ẽ contains C H (Λ) O. Hence, a concave end neighborhood is contained in an mc-p-endneighborhood. (ii) Let V be an mc-p-end neighborhood of Ẽ. Then define S to be the set of endpoints in Cl( O) of maximal segments in V from vẽ in directions of ΣẼ. Then S is diffeomorphic to ΣẼ by the map induced by radial segments as shown in the paragraph before. Thus, S/π 1 (Ẽ) is a compact set since S is contractible and ΣẼ/π 1 (Ẽ) is a K(π 1 (Ẽ))-space up to taking a torsion-free finite-index subgroup by Theorem 2.11 (Selberg s lemma). We can approximate S in the do -sense by smooth convex boundary component S ε outwards of a generalized CA-lens since Ẽ satisfies the uniform middle-eigenvalue condition: we can see this from the fact that the strict convexity in the smooth category is preserved under small perturbations. A component U S ε is a concave p-end neighborhood. (ii) follows from this. (iii) Since a concave p-end neighborhood is a proper p-end neighborhood by Theorems 6.30(iv) and 6.31, and V is a union of concave p-end neighborhoods, we obtain g(v ) V = /0 or g(v ) = V for g π 1 (O) by (ii). Suppose that g(cl(v ) O) Cl(V ) /0. Then g(v ) = V and g π 1 (Ẽ): Otherwise, g(v ) V = /0, and g(cl(v ) O) meets Cl(V ) in a totally geodesic hypersurface S equal to C H (Λ) o by the concavity of V. Furthermore, for every g π 1 (O), g(s) = S, since S is a maximal totally geodesic hypersurface in O. Hence, g(v ) S V = O since these are subsets of a properly convex domain O, the boundary of V and g(v ) are in S, and S is now in the interior of O. Then π 1 (O) acts on S, and S/G is homotopy equivalent to O/G for a finite-index torsion-free subgroup G of π 1 (O) by Theorem 2.11 (Selberg s lemma). This contradicts the condition (IE). Hence, only possibility is that Cl(V ) O = V S for a hypersurface S and g(v S) V S = /0 or g(v S) = V S for g π 1 (O). Now suppose that S bdo /0. Let S be a maximal totally geodesic domain in Cl(V ) containing S. Then S bdo by convexity and Lemma 2.46, meaning that S = S bdo. In this case, O is a cone over S and the end vertex vẽ of Ẽ. For each g π 1 (O), g(v ) V /0 meaning g(v ) = V since g(vẽ) is on Cl(S). Thus, π 1 (O) = π 1 (Ẽ). This contradicts the infinite index condition of π 1 (Ẽ). We showed that Cl(V ) O = V S for a hypersurface S and covers a submanifold in O which is a closure of an end-neighborhood covered by V. Thus, an mc-p-end-neighborhood Cl(V ) O is a proper end neighborhood of Ẽ with compact embedded boundary S/π 1 (Ẽ). Therefore we can choose positive ε so that an ε-mc-p-end-neighborhood is a proper p-end neighborhood also. This proves (iv). (v) For two mc-p-end neighborhoods U and V for different R-p-ends, we have U V = /0 by a reasoning as in (iii) replacing g(v ) with U: We showed that Cl(V ) O for an mc-p-end-neighborhood V covers an end neighborhood in O. Suppose that U is another mc-p-end neighborhood different from V. We claim that Cl(U) Cl(V ) O = /0: Suppose not. g(cl(v )) for g ΓẼ must be a subset of U since otherwise we have a situation of (iii) for V and g(v ). Since the preimage of the end neighborhoods are disjoint, g(v ) is a p-end neighborhood of the same end as

175 APPLICATIONS U. Since both are ε-mc-p-end-neighborhood which are canonically defined, we obtain U = g(v ). This was ruled out in (iii). [SS n ] 7.3. The strong irreducibility of the real projective orbifolds. The main purpose of this section is to prove Theorem 1.21, the strong irreducibility result. In particular, we don t assume the holonomy group of π 1 (O) is strongly irreducible for results from now on. But we will discuss the convex hull of the ends first. We show that the closure of convex hulls of p-end neighborhoods are disjoint in bdo. The infinity of the number of these will show the strong irreducibility. For the following, we need a stronger condition of lens-shaped ends, and not just the generalized lens-shaped property, to obtain the disjointedness of the closures of p-end neighborhoods. COROLLARY Let O be a strongly tame properly convex real projective orbifold with lens-shaped or horospherical ends and satisfy (IE) and (NA). Let U be the collection of the components of the inverse image in O of the union of disjoint collection of end neighborhoods of O. Now replace each of the p-end neighborhoods of radial lens-shaped of collection U by a concave p-end neighborhood by Corollary 7.13 (iii). Then the following statements hold : (i) Given horospherical, concave, or one-sided lens p-end-neighborhoods U 1 and U 2 contained in U, we have U 1 U 2 = /0 or U 1 = U 2. (ii) Let U 1 and U 2 be in U. Then Cl(U 1 ) Cl(U 2 ) bdo = /0 or U 1 = U 2 holds. PROOF. Suppose first that O S n. (i) Suppose that U 1 and U 2 are p-end neighborhoods of R-p-ends. Let U 1 be the interior of the associated generalized lens-cone of U 1 in Cl( O) and U 2 be that of U 2. Let U i be the concave p-end-neighborhood of U i for i = 1,2 by Corollary 7.13 (iii) that cover respective end neighborhoods in O Since the neighborhoods in U are mutually disjoint, Cl(U 1 ) Cl(U 2 ) O = /0 or U 1 = U 2. (ii) Assume that U i U, i = 1,2, and U 1 U 2. Suppose that the closures of U 1 and U 2 intersect in bd O. Suppose that they are both R-p-end neighborhoods. Then the respective closures of convex hulls I 1 and I 2 as obtained by Proposition 7.11 intersect as well. Take a point z Cl(U 1 ) Cl(U 2 ) bd O. Let p 1 and p 2 be the respective p-end vertices of U 1 and U 2. We assume that p 1 p o 2 O. Then p 1 z S(p 1 ) and p 2 z S(p 2 ) and these segments are maximal since otherwise U 1 U 2 /0. The segments intersect transversally at z since otherwise we violated the maximality in Theorems 6.30 and We obtain a triangle (p 1 p 2 z) in Cl( O) with vertices p 1, p 2,z. Suppose now that p 1 p o 2 bdo. We need to perturb p 1 and p 2 inside bdo by a small amount so that p 1 p 2 O. Let P be the 2-dimensional plane containing p 1, p 2,z. Consider a disk P Cl( O) containing p 1, p 2,z in the boundary. However, the disk has an angle π at z since Cl( O) is properly convex. We will denote the disk by (p 1 p 2 z) and p 1, p 2,z are considered as vertices. We define a convex curve α i := (p 1 p 2 z) S I i with an endpoint z for each i, i = 1,2. Let Ẽ i denote the R-p-end corresponding to p i. Since α i maps to a geodesic in R pi ( O), there exists a foliation T of (p 1 p 2 z) by maximal segments from the vertex p 1. There is a natural parametrization of the space of leaves by R as the space is projectively equivalent to an open interval using the Hilbert metric of the interval. We parameterize α i by these

176 7.3. THE STRONG IRREDUCIBILITY OF THE REAL PROJECTIVE ORBIFOLDS. 161 p 1 2 β(t ) i p α 1 β (t ) i+1 α 2 z FIGURE 2. The diagram of the quadrilateral bounded by β(t i ),β(t i+1 ),α 1,α 2. parameters as α i intersected with a leaf is a unique point. They give the geodesic length parameterizations under the Hilbert metric of R pi ( O) for i = 1,2. We now show that an infinite-order element of π 1 (Ẽ 1 ) is the same as one in π 1 (Ẽ 2 ): By convexity of I 1 and α 2, either α 2 goes into I 1 and not leave again or α 2 is disjoint from I 1. Suppose that α 2 goes into I 1 and not leave it again. Since Ẽ 2 is an R-p-end of lens-type and not just generalized lens-type, S I 2 /π 1 (Ẽ 2 ) is a compact orbifold in O by Proposition 7.11(i). There is a sequence t i so that α 2 (t i ) converges to a point of S I 2 /π 1 (Ẽ 2 ), and the projection of α 2 (t i ) to R p1 (Ẽ 1 ) converges to a point of R p1 (Ẽ 1 )/π 1 (Ẽ 1 ). Hence, by taking a short path between α 2 (t i )s, there exists an essential closed curve c 2 in S I 2 /π 1 (Ẽ 2 ) homotopic to an element of π 1 (Ẽ 1 ) since c 2 is in a lens-cone end neighborhood of the end corresponding to Ẽ 1 under the covering map O O. This contradicts (NA). (The element can be assumed to be infinite order since we can take a finite cover of O so that π 1 (O) is torsion-free by Theorem 2.11 (Selberg s lemma).) Suppose now that α 2 is disjoint from I 1. We may assume that α 1 is disjoint from I 2 without loss of generality by switching 1 and 2. Then α1 o and αo 2 now must be both in T o O. Then α 1 and α 2 have the same endpoint z and by the convexity of α 2. We parameterize α i so that α 1 (t) and α 2 (t) are on a line segment containing α 1 (t)α 2 (t) in the triangle with endpoints in zp 1 and zp 2. We obtain d O (α 2 (t),α 1 (t)) C for a uniform constant C: We define β(t) := α 2 (t)α 1 (t)). Let γ(t) denote the full extension of β(t) in (p 1 p 2 z). One can project to the space of lines through z, a one-dimensional projective space. Then the image of β(t) are so that the image of β(t ) is contained in that of β(t) if t < t. Also, the image of γ(t) contains that of γ(t ) if t < t. Thus, the convexity of the boundary S I 1 and S I 2 shows that that the Hilbert-metric length of the segment β(t) is bounded above by the uniform constant. We have a sequence t i so that p O α 2 (t i ) x,d O (p O α 2 (t i+1 ), p O α 2 (t i )) 0,x O.

177 APPLICATIONS So we obtain a closed curve c 2,i in O obtained by taking a short path jumping between the two points. By taking a subsequence, the image of β(t i ) in O geometrically converges to a segment of Hilbert-length C. As i, we have d O (p O α 1 (t i ), p O α 1 (t i+1 )) 0 by extracting a subsequence. There exists a closed curve c 1,i in O again by taking a short jumping path. We see that c 1,i and c 2,i are homotopic in O since we can use the image of the disk in the quadrilateral bounded by α 2 (t i )α 2 (t i+1 ),α 1 (t i )α 1 (t i+1 ),β(t i ),β(t i+1 ) and the connecting thin strips between the images of β ti and β ti+1 in O. This again contradicts (NA). Now, consider when U 1 is a one-sided lens-neighborhood of a T-p-end and let U 2 be a concave R-p-end neighborhood of an R-p-end of O. Let z be the intersection point in Cl(U 1 ) Cl(U 2 ). We can use the same reasoning as above by choosing any p 1 in ΣẼ1 so that p 1 z passes the interior of Ẽ 1. Let p 2 be the R-p-end vertex of U 2. Now we obtain the triangle with vertices p 1, p 2, and z as above. Then the arguments are analogous and obtain infinite order elements in π 1 (Ẽ 1 ) π 1 (Ẽ 2 ). Next, consider when U 1 and U 2 are one-sided lens-neighborhoods of T-p-ends respectively. Using the intersection point z of Cl(U 1 ) Cl(U 2 ) O and we choose p i in bdẽ i so that zp i passes the interior of ΣẼi for i = 1,2. Again, we obtain a triangle with vertex p 1, p 2, and z, and find a contradiction as above. We finally consider when U is a horospherical R-p-end. Since Cl(U) bdo is a unique point, (iii) of Theorem 4.7 implies the result. [S n P] We modify Theorem 6.31 by replacing some conditions. In particular, we don t assume that h(π 1 (O)) is strongly irreducible. LEMMA Let O be a strongly tame properly convex real projective orbifold and satisfy (IE) and (NA). Let Ẽ be a virtually factorable R-p-end of O of generalized lensshaped. Then there exists a totally geodesic hyperspace P on which h(π 1 (Ẽ)) acts, D := P O is a properly convex domain, D o O, D o /π 1 (Ẽ) is a compact orbifold, and PROOF. Assume first that O S n. The proof of Theorem 6.31 shows that either Cl( O) is a strict join vẽ D for a totally geodesic properly convex domain D in a hyperspace, or the conclusion of Theorem 6.31 holds. In both cases, π 1 (Ẽ) acts on a totally geodesic convex compact domain D of codimension 1. D is the intersection PẼ Cl( O) for a π 1 (Ẽ)-invariant subspace PẼ. Suppose that D o is not a subset of O. Then by Lemma 2.46, D bdo. In the former case, Cl( O) is the join vẽ D. For each g π 1 (Ẽ) satisfying g(vẽ) vẽ, we have g(d) D since g(vẽ) g(d) = vẽ D. g(d) D is a proper compact convex subset of D and g(d). Moreover, Cl( O) = vẽ g(vẽ) (D g(d)). We can continue as many times as there is a mutually distinct collection of vertices of form g(vẽ). Since this process must stop, we have a contradiction since by Condition (IE), there are infinitely many distinct end vertices of form g(vẽ) for g π 1 (O). Now, we go to the alternative D o O where D o /ΓẼ is projectively diffeomorphic to ΣẼ/ΓẼ. [SS n ]

178 7.3. THE STRONG IRREDUCIBILITY OF THE REAL PROJECTIVE ORBIFOLDS. 163 Proof of Theorem Assume that O S n first. Let h : π 1 (O) SL ± (n + 1,R) be the holonomy homomorphism. Suppose that h(π 1 (O)) is virtually reducible. Then we can choose a finite cover O 1 so that h(π 1 (O 1 )) is reducible. We denote O 1 by O for simplicity. Let S denote a proper subspace where π 1 (O) acts on. Suppose that S meets O. Then π 1 (Ẽ) acts on a properly convex open domain S O for each p-end Ẽ. Thus, (S O)/π 1 (Ẽ) is a compact orbifold homotopy equivalent to one of the end orbifold. However, S O is π 1 (Ẽ)-invariant and cocompact for any other p-end Ẽ. Hence, each p-end fundamental group π 1 (Ẽ) is virtually identical to any other p-end fundamental group. This contradicts (NA). Therefore, (7.1) K := S Cl( O) bd O, where g(k) = K for every g h(π 1 (O)). We divide into steps: (A) First, we show K /0. (B) We show K = D j or K = vẽ D j for some properly convex domain D j bdo Cl(U) for a p-end neighborhood U of Ẽ. (C) Finally g(d j ) = D j for g Γ and we use Corollary 7.17 to obtain a contradiction. (A) We show that K := Cl( O) S /0: Let Ẽ be a p-end. If Ẽ is horospherical, π(ẽ) acts on a great sphere Ŝ tangent to an end vertex. Since S is Γ-invariant, S has to be a subspace in Ŝ containing the end vertex by Theorem 4.7(iii). This implies that every horospherical p-end vertex is in S. Let p be one. Since there is no nontrivial segment in bdo containing p by Theorem 4.7(iv), p equals S Cl( O). Hence, p is Γ-invariant and Γ = ΓẼ. This contradicts the condition (IE). Suppose that Ẽ is a generalized lens-shaped R-p-end. Then by the existence of attracting subspaces of some elements of ΓẼ, we have either S passes the end vertex vẽ or there exists a subspace S containing S and vẽ that is ΓẼ-invariant. In the first case, we have S Cl( O) /0, and we are done for the step (A). In the second case, S corresponds to a subspace in S n 1 vẽ and S is a hyperspace of dimension n 1 disjoint from vẽ. Thus, Ẽ is a virtually factorable R-p-end. By Theorem 2.28, we obtain some attracting fixed points in the limit sets of π 1 (Ẽ). If S is a proper subspace, then Ẽ is factorable, and S contains the attracting fixed set of some positive bisemi-proximal g, g Γ E. The uniform middle eigenvalue condition shows that positive bisemi-proximal g has attracting fixed sets in Cl(L). Since g acts on S, we obtain S Cl(L) /0 by the uniform middle eigenvalue condition. If S is not a proper subspace, then g acts on S, and S contains the attracting fixed set of g by the uniform middle eigenvalue condition. Thus, S Cl(L) /0. If Ẽ is a lens-shaped T-p-end, we can apply a similar argument using the attracting fixed sets. Therefore, S Cl( O) is a subset K of bdo of dimk 0 and is not empty. In fact, we showed that the closure of each p-end neighborhood meets K. (B) By taking a dual orbifold if necessary, we assume without loss of generality that there exists a generalized lens-shaped R-p-end Ẽ with a radial p-end vertex vẽ. As above in (A), suppose that vẽ K. There exists g π 1 (O), g(vẽ) vẽ, and g(vẽ) K bdo since g acts on K. g(vẽ) is outside the closure of the concave p-end neighborhood of Ẽ by Corollary Since K is connected, K meets Cl(L) for the CAlens or generalized CA-lens L of Ẽ.

179 APPLICATIONS If vẽ K, then again K Cl(L) /0 as in (A) using attracting fixed sets of some elements of π 1 (Ẽ). Hence, we conclude K Cl(L) /0 for a generalized CA-lens L of Ẽ. Let Σ Ẽ denote Do from Lemma Since K bdo, K cannot contain Σ Ẽ. Thus, K Cl(Σ Ẽ ) is a proper subspace of Cl(Σ ), and Ẽ must be a virtually factorable end. Ẽ By Lemma 7.18, there exists a totally geodesic domain Σ in the CA-lens. The p-end Ẽ neighborhood of vẽ equals U vẽ := ({vẽ} Σ Ẽ )o. Since π 1 (Ẽ) acts reducibly, Cl(Σ Ẽ ) = D 1 D m, where K Cl(U vẽ ) contains a join D J := i J D i for a proper subcollection J of {1,...,m}. Moreover, K Cl(Σ Ẽ ) = D J. Since g(u vẽ ) is a p-end neighborhood of g(vẽ), we obtain g(u vẽ ) = U g(vẽ ). Since g(k) = K for g Γ, we obtain that (7.2) Lemma 7.18 implies that K g(cl(σ Ẽ )) = g(d J). U g(vẽ ) U vẽ = /0 for g π 1 (Ẽ) or U g(vẽ ) = U vẽ for g π 1 (Ẽ) by the similar properties of S(g(vẼ)) and S(vẼ) and the fact that bdu vẽ O and bdu g(vẽ ) O are totally geodesic domains. Let λ J (g) denote the (dimd J + 1)-th root of the norm of the determinant of the submatrix of g associated with D J for the unit norm matrix of g. There exists a sequence of virtually central diagonalizable elements γ i π 1 (Ẽ) by Proposition 4.4 of [16] so that γ i D J I,γ i D J c I for the complement J c := {1,2,...,m} J and λ J(γ i ) λ J c (γ i ). Since the lens-shaped ends satisfy the uniform middle eigenvalue condition by Theorem 6.31, we obtain (7.3) γ i D J I,γ i D J c I for the complement J c := {1,2,...,m} J, λ J (γ i ) λ vẽ (γ i ), λ J c(γ i) λ vẽ (γ i ) 0, λ J(γ i ) λ J c(γ i ). (See Theorem 2.49.) Since vẽ,d J K, the eigenvalue condition implies that one of the following holds: K = D J,K = vẽ D J or K = vẽ D J vẽ D J by the invariance of K under γi 1 and the fact that K Cl(Σ Ẽ ) = D J. Since K Cl( O), the third case is not possible. We obtain (7.4) K = D J or K = {vẽ} D J. (C) We will explore the two cases of (7.4). Assume the second case. Let g be an arbitrary element of π 1 (O) π 1 (Ẽ). Since D J K, we obtain g(d J ) K. Recall that U vẽ S(vẼ) o is a neighborhood of points of S(vẼ) o in Cl( O). Thus, g(u vẽ S(vẼ) o ) is a neighborhood of points of g(s(vẽ) o ). Recall that D o J is in the closure of U vẽ. If D o J meets then g(vẽ D J D J ) g(u vẽ S(vẼ) o ) g(s(vẽ) o ), U vẽ g(u vẽ ) /0, and S(vẼ) o g(s(vẽ) o ) /0

180 7.3. THE STRONG IRREDUCIBILITY OF THE REAL PROJECTIVE ORBIFOLDS. 165 since these are components of O with some totally geodesic hyperspaces removed. Hence, vẽ = g(vẽ) by Theorems 6.30 and Finally, we obtain D J = g(d J ) since K = vẽ D J = g(vẽ) g(d J ). If D o J is disjoint from g(v Ẽ D J D J ), then g(d J ) D J since K = vẽ D J and g(k) = K. Since D J and g(d J ) are intersections of a hyperspace with bdo, we obtain g(d J ) = D J. Both cases of (7.4) imply that g(d J ) = D J for g π 1 (O). This implies g(d J ) = D J for g π 1 (O). Since vẽ and g(vẽ) are not equal for g π 1 (O) π 1 (Ẽ), we obtain Cl(U 1 ) g(cl(u 1 )) /0. Corollary 7.17 gives us a contradiction. Therefore, we deduced that the h(π 1 (O))-invariant subspace S does not exist. [SS n ] Equivalence of lens-ends and generalized lens-ends for strict SPC-orbifolds. COROLLARY Suppose that O is a strongly tame strictly SPC-orbifold with generalized lens-shaped R-ends or lens-shaped T-ends or horospherical ends and satisfying the conditions (IE) and (NA). Then O satisfies the triangle condition and every generalized lens-shaped R-ends are lens-shaped R-ends. PROOF. Assume first O S n. Let Ẽ be a generalized lens-shaped p-end neighborhood of O. Let L be the generalized CA-lens so that the interior U of L vẽ is a lens p-end neighborhood. Then U L is a concave p-end neighborhood. Recall the triangle condition of Definition Given a triangle T with T bdo and T o O and T Cl(U) /0 for an R-p-end neighborhood U, by the strict convexity, each edge has to be inside a set of form Cl(V ) bdo for a p-end neighborhood V. By Corollary 7.17, the edges are all in Cl(U) bdo for a single R-p-end neighborhood U. Hence, the triangle condition is satisfied. By Theorem 6.26, Ẽ is a lens-shaped p-end. [SS n ]

181

182 CHAPTER 8 The convex but nonproperly convex and non-complete-affine radial ends In previous chapters, we classified properly convex or complete radial ends under suitable conditions. In this chapter, we will study radial ends that are convex but not properly convex nor complete affine. The main techniques are the theory of Fried and Goldman on affine manifolds, and a generalization of the work on Riemannian foliations by Molino, Carrière, and so on. We will show that these are quasi-joins of horospheres and totally geodesic radial ends. These are suitable deformations of joins of horospheres and totally geodesic radial ends. This is the most technical chapter. We begin by giving a definition Introduction DEFINITION 8.1. Let ˆK be a compact properly convex subset of dimension n i 2 in S n. Let S i be a subspace of dimension i disjoint from the subspace containing ˆK and in general position. Let v be a vertex in S i. A group G acts on ˆK, S i and v and on an open set U containing v in the boundary. There is a fibration Π K : S n S i S n i 1 with fibers equal to open (i + 1)-hemispheres with boundary S i. The set of fibering open (i + 1)- hemispheres H so that H U is a nonempty open set is a convex open set projectively diffeomorphic to the interior of {x} ˆK in S n i 1 for a point x not in ˆK. Here we require that there is a point v so that H U is an open set bounded by an ellipsoid containing v unless H U is empty, and H Cl(U) = {v} for H fibering over x. Also, U/G is required to be diffeomorphic to a compact orbifold times an interval. If a R-end E of a real projective orbifold has an end neighborhood projectively diffeomorphic to U/G with the induced radial foliation from U/G, then E is called a quasi-joined end (of a totally geodesic R-end and a horospherical end) and a corresponding R-p-end is said to be a quasi-joined R-p-end also. Also, any R-end with an end-neighborhood covered by an end-neighborhood of quasi-joined R-end is called by the same name. G now is a quasi-joined end group. We will see the example later. In this chapter, we will characterize this and other types of ends named NPNC-ends. See Proposition 8.27 and Remark 8.28 for detailed understanding of quasi-joined ends Main results. Let Ẽ be an NPNC-end. Recall from Chapter 4 that the universal cover ΣẼ of the end orbifold ΣẼ is foliated by complete affine i 0 -dimensional totally geodesic leaves for i 0 > 1. The end fundamental group π 1 (Ẽ) acts on a properly convex domain K that is the space of i 0 -dimensional totally geodesic leaves foliating ΣẼ. The main result of this chapter is the following. We need the proper convexity of O. 167

183 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS THEOREM 8.2. Let O be a strongly tame properly convex real projective orbifold with radial or totally geodesic ends. Assume that the holonomy group of O is strongly irreducible. Let Ẽ be an NPNC R-p-end. The leaf space naturally identifies with the interior of a compact convex set K. Suppose that the end fundamental group satisfies the property (NS) or dimk o = 0,1 for the leaf space K o of Ẽ. We assume that the p-end holonomy group h(π 1 (Ẽ)) satisfies the transverse weak middle-eigenvalue condition for NPNC-ends with respect to a p-end vertex vẽ. Then Ẽ is of quasi-joined type R-p-end for vẽ. See Definition 8.7 for the transverse weak middle-eigenvalue condition for NPNCends. Without this condition, we doubt that we can obtain this type of results. However, it is open to investigations. In this case, Ẽ does not satisfy the uniform middle-eigenvalue condition as stated in Chapter 4 for properly convex ends. We remark that Cooper and Leitner have classified the ends when the end fundamental group is nilpotent. (See Leitner [108], [107] and [109].) Also, Ballas [5] and [4] has found some examples of quasi-joined ends when the semisimple parts are diagonal groups. Our quasi-joined ends are also classified by [6] when the holonomy group is nilpotent. Recall the dual orbifold O given a properly convex real projective orbifold O. (See Section ) The set of ends of O is in a one-to-one correspondence with the set of ends of O. We show that a dual of a quasi-joined NPNC R-p-end is a quasi-joined NPNC R-p-end. COROLLARY 8.3. Let O be a strongly tame properly convex real projective orbifold with radial or totally geodesic ends. Assume that the holonomy group of O is strongly irreducible. Let Ẽ be a quasi-joined NPNC R-p-end for an end E of O satisfying the transverse weak middle-eigenvalue condition with respect to the p-end vertex. Suppose that the end fundamental group satisfies the property (NS) or dimk o = 0,1 for the leaf space K o of Ẽ. Let O denote the dual real projective orbifold of O. Let Ẽ be a p-end corresponding to a dual end of E. Then Ẽ has a p-end neighborhood of a quasi-joined type R-p-end for a choice of a p-end vertex. In short, we are saying that Ẽ can be considered a quasi-joined type R-p-end by choosing its p-end vertex well. However, this does involve artificially introducing a radial foliation structure in an end neighborhood. We mention that the choice of the p-end vertex is uniquely determined for Ẽ to be quasi-joined Outline. In Section 8.2, we discuss the R-ends that are NPNC. First, we show that the end holonomy group h(π 1 (Ẽ)) for an end E will have an exact sequence 1 N h(π 1 (Ẽ)) N K 1 where N K is in the projective automorphism group Aut(K) of a properly convex compact set K, and N is the normal subgroup mapped to the trivial automorphism of K and K o /N K is compact. We show that ΣẼ is foliated by complete affine subspaces of dimension 1. We will explain the main eigenvalue estimates following from the transverse weak middle eigenvalue condition for NPNC-ends. Then we will explain our plan to prove Theorem 8.2. In Section 8.3, we introduce the example of joining of horospherical and totally geodesic R-ends. We will now study a bit more general situation introducing Hypothesis By computations involving the normalization conditions, we show that the above exact sequence is virtually split, and we can surprisingly show that the R-p-ends are of joined or quasi-joined types. Then we show using the irreducibility of the holonomy group of π 1 (O)

184 8.2. THE TRANSVERSE WEAK MIDDLE EIGENVALUE CONDITIONS FOR NPNC ENDS 169 that they can only be of quasi-joined type using the irreducibility. As a final part of this section in Section 8.3.3, we discuss the case when N K is discrete. We prove Theorem 8.2 for this case. In Section 8.4, we discuss when N K is not discrete. There is a foliation by complete affine subspaces as above. We use some estimates on eigenvalues to show that each leaf is of polynomial growth. The leaf closures are suborbifolds V l by the theory of Carrière [28] and Molino [119] on Riemannian foliations. They form the fibration with compact fibers. π 1 (V l ) is solvable using the work of Carrière [28]. One can then take the syndetic closure to obtain a bigger group that act transitively on each leaf following Fried and Goldman [70]. We find a unipotent cusp group acting on each leaf transitively normalized by ΓẼ. Then we show that the end also splits virtually using the theory of Section 8.3. This proves Theorem 8.2. In Section 8.5, we prove Corollary 8.3 on the duals of NPNC-ends and classifying complete ends that are not cusps. REMARK 8.4. Note that the results are stated in the space S n or RP n. Often the result for S n implies the result for RP n. In this case, we only prove for S n. In other cases, we can easily modify the S n -version proof to one for the RP n -version proof The transverse weak middle eigenvalue conditions for NPNC ends We will now study the ends where the transverse real projective structures are not properly convex but not projectively diffeomorphic to a complete affine subspace. Let Ẽ be an R-p-end of O, and let U the corresponding p-end-neighborhood in O with the p-end vertex vẽ. Let ΣẼ denote the universal cover of the p-end orbifold ΣẼ as a domain in S n 1 vẽ. In Section 8.2.1, we will discuss the general setting that the NPNC-ends satisfy. In Section 8.2.2, we will give a plan to prove Theorem 8.2. This will be accomplished in Sections 8.3 and General setting. The closure Cl( ΣẼ) contains a great (i 0 1)-dimensional sphere S i 0 1, and the convex open domain ΣẼ is foliated by i 0 -dimensional hemispheres with this boundary S i 0 1 by Proposition 2.8. (These follow from Section 1.4 of [29]. See also [63].) Let S i 0 1 denote the great (i 0 1)-dimensional sphere in S n 1 vẽ of ΣẼ. The space of i 0 -dimensional hemispheres in S n 1 vẽ The projection (8.1) with boundary S i 0 1 form a projective sphere S n i 0 1. ˆΠ K : S n 1 vẽ S i 0 1 S n i 0 1 ΣẼ K o gives us an image of ΣẼ that is the interior K o of a properly convex compact set K. Let S i 0 be a great i 0 -dimensional sphere in S n containing vẽ corresponding to the directions of S i 0 1 from vẽ. The space of (i 0 + 1)-dimensional hemispheres in S n with boundary S i 0 again has the structure of the projective sphere S n i0 1, identifiable with the above one. Each i 0 -dimensional hemisphere H i 0 in S n 1 vẽ with bdh i 0 = S i 0 1 corresponds to an (i 0 + 1)-dimensional hemisphere H i0+1 in S n with common boundary S i 0 that contains vẽ.

185 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS There is also a map (8.2) Π K : S n S i 0 S n i 0 1 U K o since S i 0 1 corresponds to S i 0 in the projection S n {vẽ,vẽ } S n 1. Let SL ± (n + 1,R) i S 0 denote the subgroup of Aut(S n ) acting on S i 0 and vẽ. The,vẼ projection Π K induces a homomorphism Π K : SL ± (n + 1,R) i S 0 SL,vẼ ± (n i 0,R). Suppose that S i 0 is h(π 1 (Ẽ))-invariant. We let N be the subgroup of h(π 1 (Ẽ)) of elements inducing trivial actions on S n i 0 1. The above exact sequence (8.3) 1 N h(π 1 (Ẽ)) Π K N K 1 is so that the kernel normal subgroup N acts trivially on S n i 0 1 but acts on each hemisphere with boundary equal to S i 0 and N K acts faithfully by the action induced from Π K. Since K is a properly convex domain, K o admits a Hilbert metric d K and Aut(K) is a subgroup of isometries of K o. Here N K is a subgroup of the group Aut(K) of the group of projective automorphisms of K, and N K is called the semisimple quotient of h(π 1 (Ẽ)) or ΓẼ. THEOREM 8.5. Let ΣẼ be the end orbifold of an NPNC R-p-end Ẽ of a strongly tame properly convex n-orbifold O with radial or totally geodesic ends. Let O be the universal cover in S n. We consider the induced action of h(π 1 (Ẽ)) on Aut(S n 1 vẽ ) for the corresponding end vertex vẽ. Then the following hold : ΣẼ is foliated by complete affine subspaces of dimension i 0, i 0 > 0. Let K be the properly convex compact domain of dimension n i 0 1 whose interior is the space of complete affine subspaces of dimension i 0. h(π 1 (Ẽ)) fixes the great sphere S i 0 1 of dimension i 0 1 in S n 1 vẽ. There exists an exact sequence 1 N π 1 (Ẽ) Π K N K 1 where N acts trivially on quotient great sphere S n i 0 1 and N K acts faithfully on a properly convex domain K o in S n i 0 1 isometrically with respect to the Hilbert metric d K. We denote by FẼ the foliation on ΣẼ or the corresponding one in ΣẼ The main eigenvalue estimations. We denote by ΓẼ the p-end holonomy group acting on U fixing vẽ. Denote the induced foliations on ΣẼ and ΣẼ by FẼ. We recall length K (g) := inf{d K (x,g(x)) x K o },g ΓẼ. We recall Definition Let V i 0+1 denote the subspace of R n+1 corresponding to S i 0. By invariance of S i 0, if R µ (g) V i 0+1 {0} for some finite collection J, then R µ (g) V i 0+1 always contains a C-eigenvector of g.

186 8.2. THE TRANSVERSE WEAK MIDDLE EIGENVALUE CONDITIONS FOR NPNC ENDS 171 DEFINITION 8.6. Let ΣẼ be the end orbifold of a nonproperly convex R-p-end Ẽ of a strongly tame properly convex n-orbifold O. Let ΓẼ be the p-end holonomy group. We fix a choice of a Jordan decomposition of g for each g ΓẼ. Let λ Tr max(g) denote the largest norm of the eigenvalue of g ΓẼ affiliated with v 0, (( v)) S n S i 0, i.e., λ Tr max(g) := max{µ v R µ (g) V i 0+1 }, which is the maximal norm of transverse eigenvalues. Also, let λmin Tr (g) denote the smallest one affiliated with a nonzero vector v, (( v)) S n S i 0, i.e., λ Tr min (g) := min{µ v R µ(g) V i 0+1 }, which is the minimal norm of transverse eigenvalues. Let λ Si 0 max(g) be the largest of the norms of the eigenvalues of g with C-eigenvectors of form v, (( v)) S i 0 and λ Si 0 min (g) the smallest such one. We will assume that the p-end holonomy group h(π 1 (Ẽ)) satisfies the transverse weak middle eigenvalue condition for NPNC-ends: DEFINITION 8.7. Let λ Si 0 max(g) denote the largest norm of the eigenvalues of g h(π 1 (Ẽ)). Let λ vẽ (g) denote the eigenvalue of g at vẽ. The transverse weak middle eigenvalue condition with respect to vẽ or the R-p-end structure is that each element g of h(π 1 (Ẽ)) satisfies (8.4) λ Tr max(g) λ vẽ (g). Theorem A.12 somewhat justifies our approach. We do believe that the weak middle eigenvalue conditions implies the transverse ones at least for relevant cases. The following proposition is very important in this chapter and shows that λmax(g) Tr and λmin Tr (g) are true largest and smallest norms of the eigenvalues of g. We will sharpen the following to inequality in the discrete and non-discrete cases. PROPOSITION 8.8. Let ΣẼ be the end orbifold of a NPNC R-p-end Ẽ of a strongly tame properly convex n-orbifold O with radial or totally geodesic ends. Suppose that O in S n (resp. RP n ) covers O as a universal cover. Let ΓẼ be the p-end holonomy group satisfying the transverse weak middle eigenvalue condition for the R-p-end structure. Let g ΓẼ. Then the following hold. (8.5) λmax(g) Tr λ Si 0 max(g) λ vẽ (g) λ Si 0 min (g) λ Tr λ 1 (g) = λ Tr max(g) and λ n+1 = λ Tr min (g). min (g) PROOF. We first assume O S n. We may assume that g is of infinite order. Suppose that λ Si 0 max(g) > λmax(g). Tr We have λ Si 0 max(g) > λ vẽ (g) by the transverse weak uniform middle eigenvalue condition. Now, λ Tr max(g) < λ Si 0 max(g) implies that R λ Si 0 := R µ (g) max (g) µ =λ Si 0 max (g)

187 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS C 1 S j B' U FIGURE 1. The figure for the proof of Proposition 8.8. v E is a subspace of V i 0+1 and corresponds to a great sphere S j. Hence, a great sphere S j, j 0, in S i 0 is disjoint from {vẽ,vẽ }. Since vẽ S i 0 is not contained in S j, we obtain j + 1 i 0. A vector space V 1 corresponds µ <λ Si 0 R µ (g) where g has strictly smaller norm max (g) eigenvalues and is complementary to R λ Si 0. Let C 1 = S(V 1 ). The great sphere C 1 is max (g) disjoint from S j but C 1 contains vẽ. Moreover, C 1 is of complementary dimension to S j, i.e., dimc 1 = n j 1. Since C 1 is complementary to S j S i 0, C 1 contains a complementary subspace C 1 to S i 0 of dimension n i 0 1. Considering the sphere S n 1 vẽ at vẽ, it follows that C 1 goes to an n i 0 1-dimensional subspace C 1 in Sn 1 vẽ disjoint from l for any complete affine leaf l. Each complete affine leaf l of ΣẼ has the dimension i 0 and meets C 1 in Sn 1 vẽ by the dimension consideration. Hence, a small ball B in U meets C 1 in its interior. (8.6) For any (( v)) B, v R n+1, v = v 1 + v 2 where (( v 1 )) C 1 and (( v 2 )) S j. (( )) We obtain g k ((( v))) = g k ( v 1 ) + g k ( v 2 ), where we used g to represent the linear transformation of determinant ±1 as well (See Remark 1.1.) By Proposition 2.20, the action of g k as k makes the former vectors very small compared to the latter ones, i.e., g k ( v 1 ) / g k ( v 2 ) 0 as k. Hence, g k ((( v))) converges to the limit of g k ((( v 2 ))) if it exists. Now choose (( w)) in C 1 B and v,(( v)) S j. We let w 1 = (( w + ε v)) and w 2 = (( w ε v)) in B for small ε > 0. Choose a subsequence {k i } so that g k i( w 1 ) converges to a point of S n. The above estimation shows that {g k i( w 1 )} and {g k i( w 2 )} converge to an antipodal pair of points in Cl(U) respectively. This contradicts the proper convexity of U as g k (B ) U and the geometric limit is in Cl(U). Also the consideration of g 1 completes the inequality, and the second equation follows from the first one. [S n T]

188 8.3. THE GENERAL THEORY The plan of the proof of Theorem 8.2. We will show that our NPNC-ends are quasi-joined type ones; i.e., we prove Theorem 8.2 by proving Theorems 8.30 in Section 8.3 and Corollary 8.29 in Section 8.4. We divide into two case: we study first the case when N K is discrete and when N K is non-discrete. We will discuss some general results. For results in Section 8.3 except for Section we do not use a discreteness assumption on the semisimple quotient group N K. We will use Hypotheses 8.12 and 8.22 generalizing our situation. We show that ΓẼ acts as scalar times an orthogonal group on a Lie group N as realized as an real unipotent abelian group R i 0. See Lemmas 8.14 and This is done by computations and coordinate change arguments and the distal group theory of Fried [68]. We refine the matrix forms in Lemma 8.14 when µ g = 1. Here the matrices are in almost desired forms. Proposition 8.20 shows the splitting of the representation of ΓẼ. One uses the transverse weak middle eigenvalue condition to realize the compact (n i 0 1)- dimensional totally geodesic domain independent of S i 0 where ΓẼ acts on. In Section , we discuss joins and quasi-joins. The idea is to show that the join cannot occur by Propositions 2.53 and This will settle the cases of discrete N K by Theorem 8.30 in Section In Section 8.4, we will settle for the cases of non-discrete N K where we will use these methods. We remark that we can always take the finite index subgroup of ΓẼ during our proofs since Definition 8.28 is a definition given up to finite index subgroups The general theory Examples. First, we give some examples The standard quadric in R i 0+1 and the group acting on it. Let us consider an affine subspace A i 0+1 of S i 0+1 with coordinates x 0,x 1,...,x i0 +1 given by x 0 > 0. The standard quadric in A i 0+1 is given by x i0 +1 = x x 2 i 0. Clearly the group of the orthogonal maps O(i 0 ) acting on the planes given by x i0 +1 = const acts on the quadric also. Also, the group of the matrices of the form v T I i0 0 v 2 2 v 1 acts on the quadric. The group of affine transformations that acts on the quadric is exactly the Lie group generated by the above cusp group and O(i 0 ). The action is transitive and each of the stabilizer is a conjugate of O(i 0 ) by elements of the above cusp group. The proof of this fact is simply that such a group of affine transformations is conjugate into a parabolic group in the i dimensional complete hyperbolic space H where the ideal fixed point is identified with ((0,...,0,1)) S i 0+1 and with bdh tangent to bda i 0.

189 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS The group of projective automorphisms of the following form is a unipotent cusp group (8.7) N ( v) := v T I i0 1 0 T for v R i 0. v 2 2 v 1 (see [61] for details.) We can make each translation direction of generators of N in ΣẼ to be one of the standard vector. Therefore, we can find a coordinate system of V i0+2 so that the generators are of (i 0 + 2) (i 0 + 2)-matrix forms (8.8) N j := e T j I i e j 1 where ( e j ) k = δ jk a row i-vector for j = 1,...,i 0. That is, N ( v) = N (v 1 ) N (v i0 ) Example of joined ends. We first begin with examples. In the following, we will explain the joined type end. EXAMPLE 8.9. Let us consider two ends E 1, a totally geodesic R-end, with the p- end-neighborhood U 1 in the universal cover of a real projective orbifold O 1 in S n i 0 1 of dimension n i 0 1 with the p-end vertex v 1, and E 2 the p-end-neighborhood U 2, a horospherical type one, in the universal cover of a real projective orbifold O 2 of dimension i with the p-end vertex v 2. Let Γ 1 denote the projective automorphism group in Aut(S n i 0 1 ) acting on U 1 corresponding to E 1. We assume that Γ 1 acts on a great sphere S n i 0 2 S n i 0 1 disjoint from v 1. There exists a properly convex open domain K in S n i 0 2 where Γ 1 acts cocompactly but not necessarily freely. We change U 1 to be the interior of the join of K and v 1. Let Γ 2 denote the one in Aut(S i 0+1 ) acting on U 2 as a as a subgroup of a cusp group. We embed S n i 0 1 and S i 0+1 in S n meeting transversely at v = v 1 = v 2. We embed U 2 in S i 0+1 and Γ 2 in Aut(S n ) fixing each point of S n i 0 1. We can embed U 1 in S n i 0 1 and Γ 1 in Aut(S n ) acting on the embedded U 1 so that Γ 1 acts on S i 0 1 normalizing Γ 2 and acting on U 1. One can find some such embeddings by finding an arbitrary homomorphism ρ : Γ 1 N(Γ 2 ) for a normalizer N(Γ 2 ) of Γ 2 in Aut(S n ). We find an element ζ Aut(S n ) fixing each point of S n i 0 2 and acting on S i 0+1 as a unipotent element normalizing Γ 2 so that the corresponding matrix has only two norms of eigenvalues. Then ζ centralizes Γ 1 S n i 0 2 and normalizes Γ 2. Let U be the strict join of U 1 and U 2, a properly convex domain. U/ Γ 1, Γ 2,ζ gives us an R-p-end of dimension n diffeomorphic to Σ E1 Σ E2 S 1 R and the transverse real projective manifold is diffeomorphic to Σ E1 Σ E2 S 1. We call the results the joined end and the joined endneighborhoods. Those ends with end-neighborhoods finitely covered by these are also called a joined end. The generated group Γ 1, Γ 2,ζ is called a joined group. Now we generalize this construction slightly: Suppose that Γ 1 and Γ 2 are Lie groups and they have compact stabilizers at points of U 1 and U 2 respectively, and we have a parameter of ζ t for t R centralizing Γ 1 S n i 0 2 and normalizing Γ 2 and restricting to a unipotent action on S i 0 acting on U 2. The other conditions remain the same. We obtain

190 8.3. THE GENERAL THEORY 175 a joined homogeneous action of the semisimple and cusp actions. Let U be the properly convex open subset obtained as above as a join of U 1 and U 2. Let G denote the constructed Lie group by taking the embeddings of Γ 1 and Γ 2 as above. G also has a compact stabilizer on U. Given a discrete cocompact subgroup of G, we obtained a p-end-neighborhood of a joined p-end by taking the quotient of U. An end with an end-neighborhood finitely covered by such a one are also called a joined end. REMARK Later we will show this case cannot occur. We will modify this construction to a construction of quasi-joined ends to be defined in Definition 8.1. This will be done by adding some translations appropriately. We continue the above example to a more specific situation. EXAMPLE Let N be as in (8.16). In fact, we let c i = 0 to simplify arguments for i = 1,...,i 0 + 1, and let N be a nilpotent group in conjugate to SO(i 0 + 1,1) acting on an i 0 -dimensional ellipsoid in S i 0+1. We find a closed properly convex real projective orbifold Σ of dimension n i 0 2 and find a homomorphism from π 1 (Σ) to a subgroup of Aut(S i 0+1 ) normalizing N. (We will use a trivial one to begin with. ) Using this, we obtain a group Γ so that 1 N Γ π 1 (Σ) 1. Actually, we assume that this is split, i.e., π 1 (Σ) acts trivially on N. We now consider an example where i 0 = 1. Let N be 1-dimensional and be generated by N 1 as in (8.9). I n i (8.9) N 1 := where i 0 = 1 and we set C 1 = 0. We take a discrete faithful proximal representation h : π 1 (Σ) GL(n i 0,R) acting on a convex cone C Σ in R n i 0. We define h : π 1 (Σ) GL(n + 1,R) by matrices h(g) 0 0 (8.10) h(g) := d 1 (g) a 1 (g) 0 d 2 (g) c(g) λ vẽ (g) where d 1 (g) and d 2 (g) are n i 0 -vectors and g λ vẽ (g) is a homomorphism as defined above for the p-end vertex and det h(g)a 1 (g)λ vẽ (g) = 1. h(g) (8.11) h(g 1 ) := 1 h(g) 1 a 1 (g) 0. d 1 (g) a 1 (g) c(g) d 1 (g) a 1 (g)λ vẽ (g) + d 2 (g) λ vẽ (g) c(g) a 1 (g)λ vẽ (g) 1 λ vẽ (g)

191 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS Then the conjugation of N 1 by h(g) gives us I n i0 0 0 (8.12) 0 a 1 (g) h(g) λ vẽ (g) a 1 (g) 1 Our condition on the form of N 1 shows that (0,0,...,0,1) has to be a common eigenvector by h(π 1 (Ẽ)), and we also assume that a 1 (g) = λ vẽ (g). The last row of h(g) equals ( 0,λ vẽ (g)). Thus, the semisimple part of h(π 1 (Ẽ)) is reducible. Some further computations show that we can take any with matrices of form (8.13) h(g) := h : π 1 (Ẽ) SL(n i 0,R). S n i0 1(g) λ vẽ (g) λ vẽ (g) λ vẽ (g) for g π 1 (Ẽ) N by a choice of coordinates by the semisimple property of the (n i 0 ) (n i 0 )-upper left part of h(g). (Of course, these are not all the examples we wish to consider but we will modify later to quasi-joined ends.) Since h(π 1 (Ẽ)) has a common eigenvector, Theorem 1.1 of Benoist [16] shows that the open convex domain K that is the image of Π K is reducible. We assume that N K = N K Z for another subgroup N K, and the image of the homomorphism g N K S n i 0 1(g) gives a discrete projective automorphism group acting properly discontinuously on a properly convex subset K in S n i0 2 with a compact quotient. Let E be the one-dimensional ellipsoid where lower right 3 3-matrix of N K acts on. From this, the end is of the join form K o /N K S1 E /Z by taking a double cover if necessary and π 1 (Ẽ) is isomorphic to N K Z Z up to taking an index two subgroups. We can think of this as the join of K o /N K with E /Z as K and E are on disjoint complementary projective spaces of respective dimensions n 3 and 2 to be denoted S(K ) and S(E ) respectively Hypotheses to derive the splitting result. These hypotheses will help us to obtain the splitting. Afterwards, we will show the NPNC-ends with transverse weak middle eigenvalue conditions will satisfy these. We will outline this subsection. In Section , we will introduce a standard coordinate system to work on, where we introduce the unipotent cusp group N = R i 0 to work with. ΓẼ normalizes N by the hypothesis. Similarity Lemma 8.14 shows that the conjugation in N by an element of ΓẼ acts as a similarity, a simple consequence of the normalization property. We use this similarity and the Benoist theory [16] to prove K-is-acone Lemma 8.16 that K decomposes into a cone {k} K where N has a nice expression for the adopted coordinates. (If an orthogonal group acts cocompactly on an open manifold, then the manifold is zero-dimensional.) In Section , splitting Proposition 8.20 shows that the end holonomy group splits. To do that we find a sequence of elements of the virtual center expanding neighborhoods of a copy of K. Here, we explicitly find a part corresponding to K bdo explicitly and k is realized by an (i 0 + 1)-dimensional hemisphere where N acts on.

192 8.3. THE GENERAL THEORY The matrix form of the end holonomy group. Let ΓẼ be an R-p-end holonomy group, and let l S n 1 vẽ be a complete i 0 -dimensional leaf in ΣẼ. Then a great sphere S i 0+1 l contains the great segments from vẽ in the direction of l. Let V i0+1 denote the subspace corresponding to S i 0 containing vẽ, and V i0+2 the subspace corresponding to S i 0+1 l. We choose the coordinate system so that vẽ = ((0,,0,1)) }{{} n+1 and points of V i 0+1 and those of V i 0+2 respectively correspond to n i {}} 0 { 0,...,0,,,, n i 0 1 {}}{ 0,...,0,,,. Since S i 0 and vẽ are g-invariant, g, g ΓẼ, is of standard form (8.14) S(g) s 1 (g) 0 0 s 2 (g) a 1 (g) 0 0 C 1 (g) a 4 (g) A 5 (g) 0 c 2 (g) a 7 (g) a 8 (g) a 9 (g) where S(g) is an (n i 0 1) (n i 0 1)-matrix and s 1 (g) is an (n i 0 1)-column vector, s 2 (g) and c 2 (g) are (n i 0 1)-row vectors, C 1 (g) is an i 0 (n i 0 1)-matrix, a 4 (g) is an i 0 -column vectors, A 5 (g) is an i 0 i 0 -matrix, a 8 (g) is an i 0 -row vector, and a 1 (g),a 7 (g), and a 9 (g) are scalars. Denote ( ) S(g) s 1 (g) Ŝ(g) =, s 2 (g) a 1 (g) and is called a semisimple part of g. Let N be a unipotent group acting on S i 0 and inducing I on S n i 0 1 also restricting to a cusp group for at least one great (i 0 + 1)-dimensional sphere S i 0+1 containing S i 0. We can write each element g N as an (n + 1) (n + 1)-matrix (8.15) I n i C g U g where C g > 0 is an (i 0 +1) (n i 0 1)-matrix, U g is a unipotent (i 0 +1) (i 0 +1)-matrix, 0 indicates various zero row or column vectors, 0 denotes the zero row-vector of dimension n i 0 1, and I n i0 1 is the (n i 0 1) (n i 0 1)-identity matrix. This follows since g acts trivially on R n+1 /V i 0+1 and g acts as a cusp group element on the subspace V i 0+2.

193 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS For v R i 0, we define I n i c 1 ( v) v (8.16) N ( v) := c 2 ( v) v c i0 +1( v) 2 v 2 v 1 v where v is the norm of v = (v 1,,v i ) R i 0. We assume that N := {N ( v) v R i 0 } is a group, which must be nilpotent. The elements of our nilpotent group N are of this form since N ( v) is the product i 0 j=1 N (e j ) v j. By the way we defined this, for each k, k = 1,...,i 0, c k : R i 0 R n i 0 1 are linear functions of v defined as c k ( v) = i 0 c k j v j for v = (v 1,v 2,...,v i0 ) j=1 so that we form a group. (We do not need the property of c i0 +1 at the moment.) From now on, we denote by C 1 ( v) the (n i 0 1) i 0 -matrix given by the matrix with rows c j ( v) for j = 1,...,i 0 and by c 2 ( v) the row (n i 0 1)-vector c i0 +1( v). The lower-right (i 0 + 2) (i 0 + 2)-matrix is form is called the standard cusp matrix form. We denote by  the matrix I n i (8.17) 0 0 A for A an i 0 i 0 -matrix. Denote by the group of form by Ô(n,i 0 ) {Ô 5 O 5 O(n,i 0 )}. Provided C 1 ( v) = 0,c 2 ( v) = 0 for all v, we call N the standard cusp group (of type (n + 1,i 0 )). The standard parabolic group (of type (n + 1,i 0 )) is N Ô(n,i 0 ). The assumptions for this subsection are as follows: We will assume that the group satisfies the condition virtually only since this will be sufficient for our purposes. HYPOTHESIS Let K be defined as above for an R-p-end Ẽ. Assume that K o /N K is a compact set. (i.e., N K acts on K o in a sweeping manner.) N K acts on each factor of K = K 1 K m for the maximal decomposition of K. ΓẼ satisfies the transverse weak middle eigenvalue condition for the R-p-end structure. And elements are in the matrix form of (8.14) under a common coordinate system. A group N of form (8.16) in this coordinate system acts on each hemisphere with boundary S i, and fixes vẽ S i.

194 8.3. THE GENERAL THEORY 179 N N. The p-end holonomy group ΓẼ normalizes N. N acts on a p-end neighborhood U of Ẽ, and acts on U S i 0+1 for each great sphere S i 0+1 containing S i 0 whenever U S i 0+1 /0. N acts on the space of i 0 -dimensional leaves of ΣẼ by an induced action. Let U be a p-end neighborhood of Ẽ. Let l be an i 0 -dimensional leaf of ΣẼ. The consideration of the projection Π K shows us that the leaf l corresponds to a hemisphere where H i 0+1 l (8.18) U l := (H i 0+1 l S i 0 ) U /0 holds. LEMMA 8.13 (Cusp). Assume Hypothesis Let l be an i 0 -dimensional leaf of ΣẼ. Let H i 0+1 l denote the i dimensional hemisphere with boundary S i 0 corresponding to l. Then N acts transitively on bdu l for U l = U H l bounded by an ellipsoid in a component of H i 0+1 l S i 0. PROOF. Since l is an i dimensional leaf of ΣẼ, we obtain H i 0 l U /0. Let J l := H i 0+1 l U /0 where N acts on. Now, l corresponds to an interior point of K. We need to change coordinates of S n i0 1 so that l goes to ((0,0,...,1)) under Π K. This involves the coordinate changes of the first n i 0 coordinates. Now, we can restrict g to H i 0+1 l so that the matrix form is truly what acts on A l. Using (8.16) and the fact that c i,i = 1,...,i 0 are linear on v, we obtain that each g N then has the form in H i 0+1 l as L( v T ) I i0 0 κ( v) v 1 since the S i 0 -part, i.e., the last i coordinates, is not changed from one for (8.16) where L : R i 0 R i 0 is a linear map. The linearity of L is the consequence of the group property. κ : R i 0 R is some function. We consider L as an i 0 i 0 -matrix. Suppose that there exists a kernel K 1 of L. We use t v K 1 {O}. As t, each orbit of N (K 1 ) N in J l stay in a hyperspace H not containing vẽ. Since H does not contain vẽ, and J l is an compact ellipsoid Ê with vẽ removed, J l H = Ê H is compact. The compactness of J l H contradicts the properness condition of the action of N in Hypothesis Also, since N is abelian, the computations of N (v)n (w) = N (w)n (v) shows that vl w T = wl v T for every pair of vectors v and w in R i 0. Thus, L is a symmetric matrix. We may obtain new coordinates x n i0 +1,...,x n by taking linear combinations of these. Since L hence is nonsingular, we can find new coordinates x n i0 +1,...,x n so that N is now

195 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS of standard form: We conjugate N by A for nonsingular A. We obtain AL v T I i0 0 κ( v) va 1 1. We thus need to solve for A 1 A 1T = L, which can be done. Now, we conjugate as we wished to. We can factorize each element of N into forms I i0 0 v T I i0 0. κ( v) v 2 v v 1 Again, by the group property, α 7 ( v) := κ( v) v 2 2 gives us a linear function α 7 : R i 0 R. Hence α 7 ( v) = κ α v for κ α R i 0. Now, we conjugate N by the matrix I i0 0 0 κ α 1 and this will put N into the standard form. Now it is clear that the orbit of N (x 0 ) for a point x 0 of J l is an ellipsoid with a point removed. as we can conjugate so that the first column entries from the second one to the (i 0 + 1)-th one equals those of the last row. This proves the transitivity of the action. Let a 5 (g) denote det(a 5 g ) 1 i 0. Define µg := a 5(g) a 1 (g) = a 9(g) a 5 (g) for g ΓẼ from Lemma LEMMA 8.14 (Similarity). Assume Hypothesis Then any element g ΓẼ induces an (i 0 i 0 )-matrix M g given by gn ( v)g 1 = N ( vm g ) where M g = 1 a 1 (g) (A 5(g)) 1 = µ g O 5 (g) 1 for O 5 (g) in a compact Lie group GẼ, and the following hold. (a 5 (g)) 2 = a 1 (g)a 9 (g) or equivalently a 5(g) a 1 (g) = a 9(g) a 5 (g). Finally, a 1 (g),a 5 (g), and a 9 (g) are all positive. PROOF. Since the conjugation by g sends elements of N to itself in a one-to-one manner, the correspondence between the set of v for N and v is one-to-one.

196 8.3. THE GENERAL THEORY 181 Since we have gn ( v) = N ( v )g for vectors v and v in R i 0 by Hypothesis 8.12, we consider (8.19) S(g) s 1 (g) 0 0 I n i s 2 (g) a 1 (g) C 1 (g) a 4 (g) A 5 (g) 0 C 1 ( v) v T I i0 0 c 2 (g) a 7 (g) a 8 (g) a 9 (g) c 2 ( v) v 2 2 v 1 where C 1 ( v) is an (n i 0 1) i 0 -matrix where each row is a linear function of v, c 2 ( v) is a (n i 0 1)-row vector, and v is an i 0 -row vector. This must equal the following matrix for some v R (8.20) I n i S(g) s 1 (g) s C 1 ( v ) v T 2 (g) a 1 (g) 0 0 I i0 0. c 2 ( v C v 2 1 (g) a 4 (g) A 5 (g) 0 ) v 1 c 2 (g) a 7 (g) a 8 (g) a 9 (g) 2 From (8.19), we compute the (4,3)-block of the result to be a 8 (g) + a 9 (g) v. From (8.20), the (4,3)-block is v A 5 (g) + a 8 (g). We obtain the relation a 9 (g) v = v A 5 (g) for every v. Since the correspondence between v and v is one-to-one, we obtain (8.21) v = a 9 (g) v(a 5 (g)) 1 for the i 0 i 0 -matrix A 5 (g) and we also infer a 9 (g) 0 and det(a 5 (g)) 0. The (3,2)- block of the result of (8.19) equals a 4 (g) + A 5 (g) v T. The (3,2)-block of the result of (8.20) equals (8.22) C 1 ( v )s 1 (g) + a 1 (g) v T + a 4 (g). Thus, (8.23) A 5 (g) v T = C 1 ( v )s 1 (g) + a 1 (g) v T. For each g, we can choose a coordinate system so that s 1 (g) = 0 as Ŝ(g) is bi-semiproximal having a fixed point in S n i0 1, which involves the coordinate changes of the first n i 0 coordinate functions only. Since N acts on S i 0+1 l for some leaf l as a cusp group by Lemma 8.13, there exists a coordinate change involving the last (i 0 + 1)-coordinates x n i0 +1,...,x n,x n+1 so that the matrix form of the lower-right (i 0 + 2) (i 0 + 2)-matrix of each element N is of the standard cusp form. This will not affect s 1 (g) = 0 as we can check from the proof of Lemma 8.13 as the change involves the above coordinates only. Denote this coordinate system by Φ g,l. Let us use Φ g,l for a while using primes for new set of coordinates functions. Now A 5 (g) is conjugate to A 5(g) as we can check in the proof of Lemma Under this

197 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS coordinate system for given g, we obtain a 1 (g) 0 and we can recompute to show that a 9 (g) v = v A 5 (g) for every v as in (8.21). By (8.23) recomputed for this case, we obtain (8.24) v = 1 a 1 (g) v(a 5 (g))t as s 1 (g) = 0 here since we are using the coordinate system Φ g,l. Since this holds for every v R i 0, we obtain a 9(g)(A 5 (g)) 1 = 1 a 1 (g)(a 5 (g))t. Hence 1 det(a 5 (g)) 1/i 0 A 5 (g) O(i 0). Also, a 9 (g) a 5 (g) = a 5 (g) a 1 (g). Here, A 5 (g) is a conjugate of the original matrix A 5(g) by linear coordinate changes as we can see from the above processes to obtain the new coordinate system. This implies that the original matrix A 5 (g) is conjugate to an orthogonal matrix multiplied by a positive scalar for every g. The set of matrices {A 5 (g) g ΓẼ} forms a group since every g is of a standard matrix form (see (8.14)). Given such a group of matrices normalized to have determinant ±1, we obtain a compact group { } 1 GẼ := A i deta 5 (g) 1 5 (g) 0 g Γ Ẽ by Lemma This group has a coordinate system where every element is orthogonal by a coordinate change of coordinates x n i0 +1,...,x n. Also, a 1 (g),a 5 (g),a 9 (g) are conjugation invariant. Hence, we proved a 9 (g) a 5 (g) = a 5(g) a 1 (g). We have a 9 (g) = λ vẽ (g) > 0. Since a 5 (g) 2 = a 1 (g)a 9 (g), we obtain a 1 (g) > 0. Finally, a 5 (g) > 0 by definition. LEMMA Suppose that G is a subgroup of a linear group GL(i 0,R) where each element is conjugate to an orthogonal element. Then G is a compact group. PROOF. Clearly, the norms of eigenvalues of g G are all 1. G is virtually an orthopotent group by Moore [120] (see also [56]). Since the group is linear and for each element g, {g n n Z} is a bounded collection of matrices, G is a subgroup of an orthogonal group under a coordinate system. v T. We denote by (C 1 ( v), v T ) the matrix obtained from C 1 ( v) by adding a column vector LEMMA 8.16 (K is a cone). Assume Hypothesis Suppose that ΓẼ acts semisimply on K o. Then the following hold: K is a cone over a totally geodesic (n i 0 2)-dimensional domain K. The rows of (C 1 ( v), v T ) are proportional to a single vector, and we can find a coordinate system where C 1 ( v) = 0 not changing any entries of the lower-right (i 0 + 2) (i 0 + 2)-submatrices for all v R i 0.

198 8.3. THE GENERAL THEORY 183 (8.25) We can find a common coordinate system where O 5 (g) 1 = O 5 (g) T,O 5 (g) O(i 0 ), s 1 (g) = s 2 (g) = 0 for all g ΓẼ. In this coordinate system, we have (8.26) a 9 (g)c 2 ( v) = c 2 (µ g vo 5 (g) 1 )S(g) + µ g vo 5 (g) 1 C 1 (g). PROOF. The assumption implies that M g = µ g O 5 (g) 1 by Lemma We consider the equation (8.27) gn ( v)g 1 = N (µ g vo 5 (g) 1 ). We change to (8.28) gn ( v) = N (µ g vo 5 (g) 1 )g. Considering the lower left (n i 0 ) (i 0 + 1)-matrix of the left side of (8.28), we obtain (8.29) C 1(g), a 4 (g) + a 5(g)O 5 (g)c 1 ( v), a 5 (g)o 5 (g) v T c 2 (g), a 7 (g) a 8 (g)c 1 ( v) + a 9 c 2 ( v), a 8 (g) v T + a 9 (g) v v/2 where the entry sizes are clear. From the right side of (8.28), we obtain C 1(µ g vo 5 (g) 1 ), µ g O 5 (g) 1,T v T Ŝ(g)+ (8.30) c 2 (µ g vo 5 (g) 1 ), v v/2 C 1 (g), a 4 (g). v C 1 (g) + c 2 (g), a 7 (g) + v a 4 (g) From the top rows of (8.29) and (8.30), we obtain that ) (a 5 (g)o 5 (g)c 1 ( v),a 5 (g)o 5 (g) v T = (8.31) ( (µ g C 1 vo5 (g) 1) ), µ g O 5 (g) 1,T v T Ŝ(g). We multiplied the both sides by O 5 (g) 1 from the right and by Ŝ(g) 1 from the left to obtain ) (a 5 (g)c 1 ( v),a 5 (g) v T Ŝ(g 1 ) = (8.32) ) (µ g O 5 (g) 1 C 1 ( vo 5 (g) 1 ), µ g O 5 (g) 1 O 5 (g) 1,T v T. Let us form the subspace V C in the dual sphere R n i0 spanned by row vectors of (C 1 ( v), v T ). Let SC denote the corresponding subspace in Sn i0 1. Then { } 1 1 Ŝ(g) g ΓẼ n i det Ŝ(g) 0 1 acts on V C as a group of bounded linear automorphisms since O 5 (g) G for a compact group G. Therefore, {Ŝ(g) g ΓẼ} on SC is in a compact group of projective automorphisms by (8.32).

199 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS We recall that the dual group NK of N K acts on the properly convex dual domain K of K in a sweeping manner by Proposition Then g acts as an element of a compact group on SC. Thus, N K is reducible. Now, we apply the theory of Vey [134] and Benoist [16]: Since NK is semisimple by above premises, NK acts on a complementary subspace of S N. By Proposition 2.62, K has an invariant subspace K1 and K 2 so that we have strict join where K = K 1 K 2 where dimk 1 = dims M,dimK 2 = dims N K 1 = K S M,K 2 = K S N. Also, NK is isomorphic to a cocompact subgroup of N K,1 N K,2 A,A R where Nk,i is the restriction image of N K to K i for i = 1,2, and NK,i acts on the interior of K i properly and cocompactly. But since NK,1 acts orthogonally on S M, the only possibility is that dims M = 0: Otherwise K 0 1 /N K is not compact contradicting Proposition Hence, dimsc = 0 and K 1 is a singleton and K 2 is n i 0 2-dimensional properly convex domain. Rows of (C 1 ( v), v T ) are elements of the 1-dimensional subspace in R n i0 1 corresponding to SC. Therefore this shows that the rows of (C 1( v), v T ) are proportional to a single row vector. Since (C 1 ( e j ), e T j ) has 0 as the last column element except for the jth one, only the jth row of C 1 ( e j ) is nonzero. Let C 1 (1, e 1 ) be the first row of C 1 ( e 1 ). Thus, each row of (C 1 ( e j ), e T j ) equals to a scalar multiple of (C 1(1, e 1 ),1) for every j. Now we can choose coordinates of R n i0 so that this (n i 0 )-row-vector now has coordinates (0,...,0,1). We can also choose so that K1 is given by setting the last coordinate be zero. With this change, we need to do conjugation by matrices with the top left (n i 0 1) (n i 0 1)-submatrix being different from I and the rest of the entries staying the same. This will not affect the expressions of matrices of lower right (i 0 + 2) (i 0 + 2)-matrices involved here. Thus, n i {}} 0 { C 1 ( v) = 0 in this coordinate for all v R i 0. Also, ((0,...,0,1)) is an eigenvector of every elements of NK. The hyperspace containing K2 is also N K -invariant. Thus, the (n i 0)-vector (0,...,0,1) corresponds to an eigenvector of every element of N K. In this coordinate system, K is a strict join of a point for an (n i 0 )-vector n i 0 {}}{ k = ((0,...,0,1)) and a domain K given by setting x n i0 = 0 in a totally geodesic sphere of dimension n i 0 2 by duality. We also obtain s 1 (g) = 0,s 2 (g) = 0. For the final item we have under our coordinate system. S(g) a (8.33) g = 1 (g) 0 0, C 1 (g) a 4 (g) a 5 (g)o 5 (g) 0 c 2 (g) a 7 (g) a 8 (g) a 9 (g)

200 8.3. THE GENERAL THEORY 185 I n i (8.34) N ( v) = 0 v T. I 0 1 c 2 ( v) 2 v 2 v 1 The normalization of N shows as in the proof of Lemma 8.14 that O 5 (g) is orthogonal now. (See (8.21) and (8.23).) By (8.27), we have gn ( v) = N ( v )g, v = µ g vo 5 (g) 1. We consider the lower-right (i 0 + 1) (n i 0 )-submatrices of gn ( v) and N ( v )g. For the first one, we obtain C 1(g), a 4 (g) + a 5(g)O 5 (g), 0 0, vt 1 c 2 (g), a 7 (g) a 8 (g), a 9 (g) c 2 ( v), 2 v 2 For N ( v )g, we obtain 0, v T c 2 ( v ), 1 2 v 2 S(g), 0 0, a 1 (g) + I, 0 v, 1 C 1(g), c 2 (g), a 4 (g) a 9 (g). Considering (2,1)-blocks, we obtain c 2 (g) + a 9 (g)c 2 ( v) = c 2 ( v )S(g) + v C 1 (g) + c 2 (g). LEMMA Assume Hypothesis Then we can find coordinates so that the following holds for all g : (8.35) (8.36) a 9 (g) a 5 (g) O 5(g) 1 a 4 (g) = a 8 (g) T or a 9(g) a 5 (g) a 4(g) T O 5 (g) = a 8 (g). If µ g = 1, then a 1 (g) = a 9 (g) = λ vẽ (g) and A 5 (g) = λ vẽ (g)o 5 (g). PROOF. Again, we use (8.19) and (8.20). We need to only consider lower right (i 0 + 2) (i 0 + 2)-matrices. a 1 (g) a 4 (g) a 5 (g)o 5 (g) 0 v T (8.37) I 0 1 a 7 (g) a 8 (g) a 9 (g) 2 v 2 v 1 a 1 (g) 0 0 (8.38) = a 4 (g) + a 5 (g)o 5 (g) v T a 5 (g)o 5 (g) 0. a 7 (g) + a 8 (g) v T + a 9(g) 2 v 2 a 8 (g) + a 9 (g) v a 9 (g)

201 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS This equals (8.39) (8.40) a v T 1 (g) 0 0 I 0 1 a 2 v 2 4 (g) a 5 (g)o 5 g 0 v 1 a 7 (g) a 8 (g) a 9 (g) a 1 (g) 0 0 = a 1 (g) v T + a 4 (g) a 5 (g)o 5 (g) 0 v 2 + v a 4 (g) + a 7 (g) a 5 (g) v O 5 (g) + a 8 (g) a 9 (g) a 1 (g) 2. Then by comparing the (3,2)-blocks, we obtain Thus, v = a 5(g) a 9 (g) v O 5 (g). From the (3,1)-blocks, we obtain a 8 (g) + a 9 (g) v = a 8 (g) + a 5 (g) v O 5 (g). a 1 (g) v v /2 + v a 4 (g) = a 8 (g) v T + a 9 (g) v v/2. Since the quadratic forms have to equal each other, we obtain a 9 (g) a 5 (g) vo 5(g) 1 a 4 (g) = v a 8 (g) for all v R i 0. Thus, a 9(g) a 5 (g) (O 5(g) T a 4 (g)) T = a 8 (g) T. Since we have µ g = 1, we obtain a 1 (g) = a 9 (g) = a 5 (g) = λ vẽ (g) and A 5 (g) = λ vẽ (g)o 5 (g) by Lemma Also, a 1 (g) = a 9 (g) = a 5 (g) = λ vẽ (g). By above, we conclude that each g ΓẼ has the form S(g) a (8.41) 1 (g) 0 0 C 1 (g) a 1 (g) v T g a 5 (g)o 5 (g) 0 c 2 (g) a 7 (g) a 5 (g) v g O 5 (g) a 9 (g). REMARK Since the matrices are of form (8.41), g µ g is a homomorphism. COROLLARY If g of form of (8.41) centralizes a Zariski dense subset A of N, then µ g = 1 and O 5 (g) = I i0. PROOF. N is isomorphic to R i 0. The subset A of R i 0 corresponding to A is also Zariski dense in R i 0. gn ( v) = N ( v)g shows that v = vo 5 (g) for all v A. Hence O 5 (g) = I Invariant α 7. We assume µ g = 1 identically in this subsubsection. When µ g = 1 for all g ΓẼ, by taking a finite index subgroup of ΓẼ, we conclude that each

202 8.3. THE GENERAL THEORY 187 g ΓẼ has the form S(g) λ (8.42) M(g) := vẽ (g) 0 0 C 1 (g) λ vẽ (g) v T. g λ vẽ (g)o 5 (g) 0 c 2 (g) a 7 (g) λ vẽ (g) v g O 5 (g) λ vẽ (g) We define an invariant: α 7 (g) := a 7(g) λ vẽ (g) v g 2. 2 We denote by ˆM(g) the lower right (i 0 + 1) (i 0 + 1)-submatrix of M(g). An easy computation shows that ˆM(g) ˆM(h) = ˆM(gh) where v gh = v + O 5 (g) v h holds. Then it is easy to show that α 7 (g n ) = nα 7 (g) and α 7 (gh) = α 7 (g) + α 7 (h), whenever g,h,gh G. We obtain a homomorphism to the additive group R α 7 : ΓẼ R. (See (8.43).) Here α 7 (g) is determined by factoring the matrix of g into commuting matrices of form I n i (8.43) 0 0 I i0 0 0 α 7 (g) 0 1 S g λ vẽ (g) 0 0. C 1 (g) λ vẽ (g) v g λ vẽ (g)o 5 (g) 0 c 2 (g) λ vẽ (g) v 2 2 λ vẽ (g) v g O 5 (g) λ vẽ (g) Splitting the NPNC end. Let Z = Z (G) for any subgroup G of SL ± (n + 1,R) denote the Zariski closure in SL ± (n + 1,R). PROPOSITION 8.20 (Splitting). Assume Hypothesis 8.12 for ΓẼ. Suppose additionally the following: Suppose that N K acts on K in a semi-simple manner. µ g = 1 for every g ΓẼ. K = {k} K a strict join, and K o /N K is compact. Then the following hold: there exists an exact sequence 1 N ΓẼ,N Π K N K 1. K embeds projectively in the closure of bd O invariant under ΓẼ, and

203 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS one can find a coordinate system so that every N ( v) for v R i 0 and each element g of ΓẼ are written so that C 1 ( v) = 0,c 2 ( v) = 0, and C 1 (g) = 0 and c 2 (g) = 0. PROOF. (A) Let Z denote ΓẼ,N. Since N N, we have homomorphism Z Π K N K 1. extending Π K of (8.3). We now determine the kernel. The function λ vẽ : ΓẼ R + extends to λ vẽ : Z R +. By (8.42), we deduce that every element g of Z is of form S(g) λ (8.44) vẽ (g) 0 0 C 1 (g) λ vẽ (g) v T. g λ vẽ (g)o 5 (g) 0 c 2 (g) a 7 (g) λ vẽ (g) v g O 5 (g) λ vẽ (g) Notice that α 7 is identical zero on N. Since N N by Hypothesis 8.12, α 7 is zero on the kernel N. Hence, g N is of form I n i (8.45) g = C 1 ( v g ) v T g I i0 0 v g 2 c 2 ( v g ) 2 v g 1 since α 7 (g) = 0, and S(g) = λ g I n i0 1 and the (n i 0,n i 0 )-term must be λ g for some λ g > 0 so that it goes to I in K. Since the kernel of Π K Z is generated by N and N, we proved the first item. (B) Since K/N K is compact, N K contains the virtual center Z of N K by Proposition We take a finite index group Γ Ẽ of ΓẼ so that the image in N K acts on each K i in the factor of K = K 1 K n by Propositions 2.62 (see also 2.49). Denote by N K the image of ΓẼ. Also, the Zariski closure Z (ΓẼ) of ΓẼ normalizes N since ΓẼ by Hypothesis 8.12 and N is an algebraic unipotent Lie group. By Proposition 2.49, we find an element g Z (Γ Ẽ ) going to an element ĝ of the center Z( N K ) so that ĝ(k ) = K and ĝ(k) = k so that the corresponding matrix in SL ± (n i 0,R) with the norm of the eigenvalue at k is > 1 and all of those at K are < 1. g is of form by Lemma 8.17 or (8.42): S(g) n i λ vẽ (g) 0 0 (8.46) C 1 (g) λ vẽ (g) v T g λ vẽ (g)o 5 (g) 0 ) c 2 (g) λ vẽ (g) (α 7 (g) + v g 2 λ vẽ (g) v g O 5 (g) λ vẽ (g) where the eigenvalues of S(g) n i0 1 have norms < 1 and λ vẽ > 1. We can characterize S(K g ) as the g-invariant subspace corresponding to subspaces associated with the sum of the real primary subspaces with eigenvalues of norms < λ vẽ. 2

204 8.3. THE GENERAL THEORY 189 We choose a coordinate system of S n so that g is of form so that C 1 (g) = 0,c 2 (g) = 0. Then a compact proper convex domain K g in S(K g ) maps to K under under the projection Π K : S n S i 0 S n i0 1. Now assume that C 1 (g) = 0,c 2 (g) = 0. (See Definition 2.19 for real primary subspace.) (C) The elements of N, N N are of form: I n i (8.47) k = C 1 ( v k ) v T. k I i0 0 v c 2 ( v k ) k 2 2 v k 1 We obtain the first (n i 0 1) columns of gn ( v k )g 1 is of form (8.48) I n i λ vẽ (g)o 5 (g)c 1 ( v k ) λ vẽ (g)( v g O 5 (g)c 1 ( v k ) + c 2 ( v k )) The other parts are identical with the rest of the matrix in (8.47) with v k replaced by v k O 5 (g) 1 but keeping v T k unchanged. Hence, gn ( v k)g 1 = N ( v k O 5 (g) 1 ). We thus need to have (8.49) λ vẽ (g)o 5 (g)c 1 ( v k ) = C 1 ( v k O 5 (g) 1 ), λ vẽ (g)( v g O 5 (g) 1 C 1 ( v k ) + c 2 ( v k )) = c 2 ( v k O 5 (g) 1 ). Since λ vẽ (g) > 1 and the first equation implies λ vẽ (g) n O 5 (g n )C 1 ( v k ) = C 1 ( v k O 5 (g n ) 1 ) for all n. Since O 5 (g n ) approximates I for some sequence n i, n i, the right side is bounded, and the left side is going to +, the equality holds only if C 1 ( v k ) = 0. Then from the second equation, we obtain c 2 ( v k ) = 0 similarly. Since g goes to the center Z(N K ) of N K under Π K, for any elements j of ΓẼ, jg j 1 g 1 = N ( v), and hence, (8.50) jg j 1 = N ( v)g for some v R i 0. Since C 1 ( v k ) = 0 and c 2 ( v k ) = 0, we have N ( v)g acts on K g, and S(K g ) = S C λi (g) R n+1 i { j λ j <1} using Definition Let j be an element of Z. Then jg j 1 acts on j(k g ), and j(s(k g )) is the span of real primary subspaces with eigenvalues of norms < 1. Since N ( v) acts trivially on S(K g ) and N ( v)g has the same set of eigenvalues associated with S(K g ) as g, (8.50) implies j(s(k g )) = S( j(k g )) = S(K g ). Since they go to K under Π K, we obtain that j(k g ) = K g. Hence, j acts on this set.

205 THE CONVEX BUT NONPROPERLY CONVEX AND NON-COMPLETE-AFFINE RADIAL ENDS We obtain that Γ Ẽ acts on K g. Let F be the compact set in K o so that N K (F) = K o since N K is sweeping. We take a segment s in K o with end points k and in the interior of K and a sequence of elements x i on it converging to k. We take g i so that g i (x i ) F. Then g i K is bounded since g i (s) is in a compact set F in K o that is the image of F. Hence, we may assume that g i K is converging to an element of Aut(K ). Thus, we can find a sequence g i in Γ Ẽ so that 1 ĝ i := det(s(g i ) n i0 1) S(g i) n i0 1 K g g Aut(K ) and the eigenvalue associated with k of ĝ i goes to zero. Hence, g i K g g and λ vẽ (g i ) 0. Since the sequence of the associated eigenvalue of K g of g i goes to, S(K g ) is in  (g i ). Theorem 2.37 shows that S(K g ) Cl(Ω) bdω. Let S i 0+1 k be the sphere that is the closure of the inverse image corresponding to {k,k } under the projection Π K. For each g i, S(K g ) and S i 0+1 k satisfy the premise of Proposition Each point x of K o has a point in U mapping to it under Π K. We have x {x } {k} for x K. Then x is a limit of {ĝ i (x)}. By the matrix form of g i, a limit argument shows that the point x of K g mapping to x in K is in Cl( O) by Proposition Hence, K g Cl( O). Since ΓẼ/Γ Ẽ is finite, we obtain finitely many sets of form g (K g ) for g ΓẼ. If they are not identical, at least one g satisfies g (K g ) K g. Then g i (g (K g )) for i Z produces infinitely many distinct sets of form g(k g ), which is a contradiction. Hence g (K g ) = K g for all g ΓẼ. This implies that C 1 (g ) = 0 and c 2 (g ) = 0 for all g ΓẼ. This proves the second item. The third item follows since C 1 ( v) = 0,c 2 ( v) = 0 for v R i 0 and ΓẼ acts on K g as we proved above. REMARK We can also prove this result with an extra condition more easily: A virtual center of ΓẼ maps to N K going to a Zariski dense group of the virtual center of Aut(K). See Remark 8.31 also. This was the earlier approach Joins and quasi-joined ends for µ 1. We will now discuss about joins and their generalizations in depth in this subsection. That is we will only consider when µ g = 1 for all g ΓẼ. We will use a hypothesis and later show that the hypothesis is true in our cases to prove the main results. Again, we assume the hypothesis virtually since it will be sufficient. HYPOTHESIS 8.22 (µ g 1). Let ΓẼ be a p-end holonomy group. We continue to assume as in Hypothesis 8.12 for ΓẼ. Every g ΓẼ M g is so that M g is in a fixed orthogonal group O(i 0 ). Thus, µ g = 1 identically. ΓẼ acts on the subspace S i 0 containing vẽ and the properly convex domain K in the subspace S n i 0 2 forming an independent pair with S i 0 mapping homeomorphic to the factor K of K = {k} K under Π K where the action is semi-simple. N acts on these two subspaces fixing every points of S n i 0 2. We choose the coordinates so that vẽ have coordinates ((0,...,0,1)). S n i 0 2 contains the standard points [e i ] for i = 1,...,n i 0 1 and S i 0+1 contains [e i ] for i = n i 0,...,n+1.

206 8.3. THE GENERAL THEORY 191 FIGURE 2. A figure of a quasi-joined R-p-end-neighborhood Let H be the open n-hemisphere defined by x n i0 > 0. Then by the convexity of ΣẼ, we can choose H so that U H o, K H and S i 0 bdh. By Hypothesis 8.22, elements of N have the form of (8.16) with C 1 ( v) = 0,c 2 ( v) = 0 for all v R i 0 and the elements of ΓẼ has the form of (8.42) with s 1 (g) = 0,s 2 (g) = 0,C 1 (g) = 0, and c 2 (g) = 0. Again we recall the projection Π K : S n S i 0 S n i 0 1. ΓẼ has an induced action on S n i 0 1 and acts on a properly convex set K in S n i 0 1 so that K equals a strict join k K for k corresponding to S i 0+1. (Recall the projection S n S i 0 to S n i 0 1. ) We recall the invariants from the form of (8.43). for every g ΓẼ. It is easy to show α 7 (g) := a 7(g) λ vẽ (g) vg 2 2 (8.51) α 7 (g n ) = nα 7 (g) and α 7 (gh) = α 7 (g) + α 7 (h), whenever g,h,gh ΓẼ. (8.52) Under Hypothesis 8.22, every g ΓẼ is of form: S g λ g λ g v T g λ g O 5 (g) 0 ( ) 0 λ g α 7 (g) + v g 2 λ g v g O 5 (g) 2 λ g,