Uniformization of 3-orbifolds

Transcription

1 Uniformization of 3-orbifolds Michel Boileau, Bernhard Leeb, Joan Porti arxiv:math/ v1 [math.gt] 18 Oct 2000 March 10, 2008 Preprint Preliminary version Abstract The purpose of this article is to give a complete written proof of the Orbifold Theorem announced by Thurston in late 1981: If O is a compact, connected, orientable, irreducible and topologically atoroidal 3-orbifold with non-empty ramification locus, then O is geometric. As a corollary, any smooth orientation preserving non-free finite group action on S 3 is conjugate to an orthogonal action. Contents Introduction 3 1 Reduction to the small case Basic definitions Proof of the Orbifold Theorem from Theorem Preliminaries on cone manifolds Main definitions Local description of cone manifolds in dimensions 2 and Cone manifolds as Alexandrov spaces The cut locus Spherical turnovers Injectivity radius Cone 3-manifolds with boundary Compactness and thickness results Geometric convergence Compactness Theorem Thickness of vertices boileau@picard.ups-tlse.fr, leeb@riemann.mathematik.uni-tuebingen.de, porti@mat.uab.es 1

2 4 Finiteness properties of cone 3-manifolds Euclidean cone 3-manifolds Hyperbolic cone 3-manifolds with cone angles < π Proof of Proposition 4.5 in case a) Proof of Proposition 4.5 in case b) Deformations of cone 3-manifolds Reduction to the case where M is hyperbolic The openness theorem Proof of the openness at t > The non-collapsing case The limit of hyperbolicity is t < the limit of hyperbolicity is t = Small 3-orbifolds with boundary The normal radius goes to zero: η( C n ) The normal radius stays away from zero: η( C n ) a > Closed small 3-orbifolds of dihedral type Coverings of small 3-orbifolds of dihedral type Reduction to the case O Σ 0 hyperbolic Increasing cone angles Collapsing sequences Non-dihedral small 3-orbifolds Spherical cone structures The variety of representations into SU(2) Holonomies and representations into SU(2) SU(2) Construction of S and of S Proof of the local rigidity theorem References 75 2

3 Introduction A 3-dimensional orientable orbifold is a metrizable space with coherent local models given by quotients of R 3 by finite subgroups of SO(3). For example, the quotient of a 3-manifold by a properly discontinuous group action naturally inherits a structure of a 3-orbifold. When the group action is finite, such an orbifold is said to be very good. The set of points having non-trivial local isotropy group is called the ramification locus of the orbifold. It is a codimension 2 trivalent graph. For a general background about orbifolds we refer to [BS1, BS2], [DaM], [Kap, Chap. 7], [Scu], [Tak1] and [Thu1, Chap. 13]. In 1981, Thurston [Thu2, Thu6] announced the proof of the Geometrization Theorem for 3-orbifolds with non-empty ramification locus, and lectured about his proof. Unfortunatly he did not publish it. Since 1986, useful notes about Thurston s proof (by Soma, Ohshika and Kojima [SOK] and by Hodgson [Ho1]) have been circulating. In 1989 much more details, when the ramification locus has no vertex (i.e. the cyclic case), appeared in Zhou s thesis [Zh1, Zh2]. The purpose of this article is to give a complete written proof of Thurston s Orbifold Theorem. A different proof has been announced by D. Cooper, C. Hodgson and S. Kerckhoff in [CHK]. The main result of this article is the following uniformization theorem, which implies Thurston s orbifold Theorem for compact orientable 3-orbifolds. Theorem 1 Let O be a compact, orientable, connected, small 3-orbifold with nonempty ramification locus, then O is geometric (i.e admits either a hyperbolic, Euclidean, spherical or Seifert fibered structure). An orientable compact 3-orbifold O is small if it is irreducible, if its boundary O is a (perhaps empty) collection of turnovers (i.e. 2-spheres with three branching points), and if it does not contain any other closed incompressible orientable 2-suborbifold. A special case of Theorem 1 concerns non-free finite group actions on the 3-sphere S 3. It recovers all the previous known partial results (cf. [DaM], [Fei], [MB], [Mor]), as well as the results about finite group actions on the 3-ball B 3 (cf. [MY2], [KS]). Corollary 2 An orientation preserving, smooth, non-free finite group action on S 3 is conjugated to an orthogonal action. Every compact orientable irreducible and atoroidal 3-orbifold can be canonically split along a maximal (perhaps empty) collection of disjoint and pairwise non-parallel hyperbolic turnovers. The pieces that are obtained are either Haken or small 3- orbifolds (cf. 1). Using an extension of Thurston s hyperbolization Theorem to the case of Haken orbifolds (cf. [BoP, Chap. 8]), we show that Theorem 1 implies the orientable case of the Orbifold Theorem: Corollary 3 (Orbifold Theorem) Let O be a compact, connected, orientable, irreducible 3-orbifold with non-empty ramification locus. If O is topologically atoroidal, then O is geometric. 3

4 F. Bonahon and L. Siebenmann s characteristic decomposition of a compact connected orientable irreducible and -irreducible 3-orbifold along toric 2-suborbifold ([BS1]) and the fact that 3-orbifolds with a geometric decomposition are very good [McCMi], imply the following corollary of the Orbifold Theorem: Corollary 4 Every compact connected orientable irreducible 3-orbifold is very good. The paper is organized as follows. In section 1 we recall some basic terminology about orbifolds. Then we deduce Thurston s Orbifold Theorem from Theorem 1. In section 2 we introduce some preliminary results on cone 3-manifolds. In section 3 we prove the Compactness Theorem, which is a cone 3-manifold version of Gromov s Compactness Theorem for Riemannian manifolds of pinched sectional curvature. Then we prove the Thick vertex Lemma, which gives useful conditions in order to apply the Compactness Theorem. In section 4, for a Euclidean cone 3-manifold with cone angles bounded away 2π we prove a finiteness result similar to the Bonnet-Myers Theorem. Then we prove the existence of a compact core for a hyperbolic cone 3-manifold of finite volume and with cone angles bounded away π. In section 5 we prove the Openness Theorem, which is the first step in the proof of Theorem 1. It shows that the deformation space of hyperbolic cone structures on a compact orientable 3-orbifold is open. The goal then is to study the limit of hyperbolicity while the cone angles are increasing. In section 6 we prove Theorem 1 in the so called non-collapsing case. That means when the orbifold angles can be reached in the deformation space of hyperbolic cone structures on the small orbifold. In section 7 we achieve the proof of Theorem 1 when the small orbifold has nonempty boundary. In section 8 we uniformize closed small orbifolds of dihedral type; this means that the singular locus is not empty and that the local isotropy group of each singular point is either cyclic or dihedral. We reduce it to the cyclic case (cf. [BoP]), by using the fact that a closed small 3-orbifold of dihedral type has a finite regular covering of cyclic type. In section 9 we uniformize closed small orbifolds of non-dihedral type. Here we use the crucial fact that hyperbolic cone structures on a non-dihedral irreducible small 3-orbifold with cone angles less than the orbifold angles may only collapse with bounded diameter. Moreover the proof for the spherical case relies on the real algebraic structure of the deformation space of spherical cone structures on the small non-dihedral orbifold with cone angles less than π, but bigger than the orbifold angles. In section 10 we prove the results needed in section 9 about the deformation space of spherical cone structures on a small 3-orbifold. 4

5 Some part of the proof of Theorem 1 (the opennness Theorem in 5 and the dihedral case in 8) presented here relies on the results and methods used in [BoP]. In particular we do not reprove the cyclic case here. In the other hand, simpler and more geometric proofs of the results on the geometry of cone 3-manifolds ( 2, 3 and 4) can be found in [BLP]. We end the introduction with a flowchart of the proof (Figures 1 and 2). We wish to thank D. Cooper for explaining us their more combinatorial approach, with C. Hodgson and S. Kerckhoff, to uniformize irreducible 3-orbifolds with finite fundamental group, and thus giving us the idea of relabelling to use the dihedral case to get the general spherical case. However, the two approaches remain quite different. 5

6 Figure 1: Flowchart (first part) 6

7 Figure 2: Flowchart of the collapsing case 7

8 1 Reduction to the small case In this section we show that Theorem 1 implies Thurston s Orbifold Theorem. 1.1 Basic definitions According to [BS1, BS2] and [Thu1, Ch.13], we use the following terminology. A compact 2-orbifold F 2 is respectively spherical, discal, toric or annular if it is the quotient by a finite smooth group action of respectively the 2-sphere S 2, the 2-disk D 2, the 2-torus T 2 or the annulus S 1 [0, 1]. A compact 2-orbifold is bad if it is not good. Such a 2-orbifold is the union of two non-isomorphic discal 2-orbifolds along their boundaries. A compact 3-orbifold O is irreducible if it does not contain any bad 2-suborbifold and if every orientable spherical 2-suborbifold bounds in O a discal 3-suborbifold, where a discal 3-orbifold is a finite quotient of the 3-ball by an orthogonal action. A connected 2-suborbifold F 2 in an orientable 3-orbifold O is compressible if either F 2 bounds a discal 3-suborbifold in O or there is a discal 2-suborbifold 2 which intersects transversally F 2 in 2 = 2 F 2 and such that 2 does not bound a discal 2-suborbifold in F 2. A 2-suborbifold F 2 in an orientable 3-orbifold O is incompressible if no connected component of F 2 is compressible in O. A properly embedded 2-suborbifold (F, F) (O, O) is -compressible if: either (F, F) is a discal 2-suborbifold (D 2, D 2 ) which is -parallel, or there is a discal 2-suborbifold O such that F is a simple arc α, M is a simple arc β, with = α β and α β = α = β. An orientable properly embedded 2-suborbifold F 2 is -parallel if it belongs to the frontier of a collar neighborhood F 2 [0, 1] O of a boundary component F 2 O. A properly embedded 2-suborbifold F 2 is essential in a compact orientable irreducible 3-orbifold, if it is incompressible, -incompressible and not -parallel. A compact 3-orbifold is topologically atoroidal if it does not contain any embedded essential orientable toric 2-suborbifold. A turnover is a 2-orbifold with underlying space a 2-sphere and ramifcation locus three points. In an irreducible orientable 3-orbifold an embedded turnover either bounds a discal 3-suborbifold or is incompressible and of non-positive Euler characteristic. An orientable compact 3-orbifold O is Haken if it is irreducible, if every embedded turnover is either compressible or -parallel, and if it contains an embedded orientable incompressible 2-suborbifold which is not a turnover. 8

9 Remark. The word Haken may lead to confusion, since it is not true that a compact orientable irreducible 3-orbifold containing an orientable incompressible properly embedded 2-suborbifold is Haken in our meaning (cf. [Dun1], [BoP, Ch. 8]). An orientable compact 3-orbifold O is small if it is irreducible, its boundary O is a (perhaps empty) collection of turnovers, and O does not contain any essential orientable 2-suborbifold. Remark. By irreduciblity, if a small orbifold O has a non-empty boundary, then either O is a discal 3-orbifold, or O is a collection of Euclidean and hyperbolic turnovers. A compact orientable 3-orbifold O is hyperbolic if its interior is orbifold-diffeomorphic to the quotient of the hyperbolic space H 3 by a non-elementary discrete group of isometries. In particular I-bundles over hyperbolic 2-orbifolds are hyperbolic, since their interiors are quotients of H 3 by non-elementary Fuchsian groups. A compact orientable 3-orbifold is Euclidean if its interior has a complete Euclidean structure. Thus, if a compact orientable and -incompressible 3-orbifold O is Euclidean, then either O is a I-bundle over a 2-dimensional Euclidean closed orbifold or O is closed. A compact orientable 3-orbifold is spherical when it is the quotient of S 3 by the orthogonal action of a finite subgroup of SO(4). A Seifert fibration on a 3-orbifold O is a partition of O into closed 1-suborbifolds (circles or intervals with silvered boundary) called fibers, such that each fiber has a saturated neighborhood diffeomorphic to S 1 D 2 /G, where G is a finite group which acts smoothly, preserves both factors, and acts orthogonally on each factor and effectively on D 2 ; moreover the fibers of the saturated neighborhood correspond to the quotients of the circles S 1 { }. On the boundary O, the local model of the Seifert fibration is S 1 D 2 +/G, where D 2 + is a half disk. A 3-orbifold that admits a Seifert fibration is called Seifert fibered. A Seifert fibered 3-orbifold which does not contain a bad 2-suborbifold is geometric (cf. [Scu], [Thu7]). Besides the constant curvature geometries E 3 and S 3, there are four other possible 3-dimensional homogeneous geometries for a Seifert fibered 3-orbifold: H 2 R, S 2 R, SL 2 (R) and Nil. The non Seifert fibered 3-orbifolds require a constant curvature geometry or Sol. Compact 3-orbifolds with Sol geometry are fibered over a closed 1-dimensional orbifold with toric fiber and thus are not topologically atoroidal (cf. [Dun2]). 1.2 Proof of the Orbifold Theorem from Theorem 1 This proof is based on the following extension of Thurston s Hyperbolization Theorem to Haken orbifolds (cf. [BoP, Chap. 8]): 9

10 Theorem 1.1 (Hyperbolization Theorem of Haken orbifolds). Let O be a compact orientable connected Haken 3-orbifold. If O is topologically atoroidal and not Seifert fibered, nor Euclidean, then O is hyperbolic. Remark 1.2 The proof of this theorem follows exactly the scheme of the proof for Haken manifolds [Thu2, Thu3, Thu5], [McM2], [Kap], [Ot1, Ot2] (cf. [BoP, Chap. 8] for a precise overview). Proof of the Orbifold Theorem (Corollary 3). Let O be a compact orientable connected irreducible topologically atoroidal 3- orbifold. By [Dun1, Thm 12] there exists in O a (possibly empty) maximal collection T of disjoint embedded pairwise non parallel essential turnovers. Since O is irreducible and topologically atoroidal, any turnover in T is hyperbolic (i.e. has a negative Euler characteristic). When T is empty, the Orbifold Theorem reduces either to Theorem 1 or to Theorem 1.1 according to whether O is small or Haken. When T is not empty, we first cut open the orbifold O along the turnovers of the family T. By maximality of the family T, the closure of each component of O T is a compact orientable irreducible topologically atoroidal 3-orbifold that does not contain any essential embedded turnover. Let O be one of these connected components. By the previous case O is either hyperbolic, Euclidean or Seifert fibered. Since, by construction, O contains at least one hyperbolic turnover T, O must be hyperbolic. Moreover any such hyperbolic turnover T in O is a Fuchsian 2-suborbifold, because there is a unique conjugacy class of faithful representations of the orbifold group π o 1(T) into PSL 2 (C). We assume first that all the connected components of O T have 3-dimensional convex cores. In this case the hyperbolic turnovers are totally geodesic boundary components of the convex cores. Hence the hyperbolic structures of the components of O T can be glued together along the hyperbolic turnovers of the family T to give a hyperbolic structure on the 3-orbifold O. If the convex core of O is 2-dimensional, then O is either a product T [0, 1], where T is a hyperbolic turnover, or a quotient of T [0, 1] by an involution. When O = T [0, 1], then the 3-orbifold O is Seifert fibered. When O is the quotient of T [0, 1], then it has only one boundary component and it is glued either to another quotient of T [0, 1] or to a component with 3- dimensional convex core. When we glue two quotients of T [0, 1] by an involution, we obtain a Seifert fibered orbifold. Finally, gluing O to a hyperbolic orbifold with totally geodesic boundary consists in identifying its faces by an isometric involution. 10

11 2 Preliminaries on cone manifolds In this section we review some results about cone manifolds. 2.1 Main definitions Definitions. For a topological space X, the cone on X is the quotient [0, ε] X/ for some ε > 0, where the relation collapses {0} X to a point, called the vertex of the cone. For a metric space X with diameter diam(x) π, the metric cone on X with curvature k R is the cone [0, ε] X/ equipped with the distance defined by the following formula: c k ( (r, x)(s, y) ) = c k (r) c k (s) + k s k (r) s k (s) cos xy when k 0, (r, x)(s, y) 2 = r 2 + s 2 2 r s cos xy when k = 0, where s k is the unique function satisfying s k +k s k = 0, s k (0) = 0 and ṡ k (0) = 1, and c k = ṡ k. In addition, when k > 0 we assume ε < π/ k. This metric cone is denoted by C k,ε (X). Remarks. We recall that c 1 = cos, s 1 = sin, c 1 = cosh, s 1 = sinh, c 0 = 1, s 0 (r) = r. The formula that defines the distance on C k,ε (X) is the cosinus formula in constant curvature k for the triangle formed by (r, x), (s, y) and the vertex of the cone (0, ). When X is a manifold with an infinitesimal metric dσx 2 defined on X minus a codimension 2 subset, then there is an infinitesimal metric defined on the metric cone [0, ε] X/ minus a codimension 2 subset as: dσ 2 = dr 2 + s 2 k(r) dσ 2 X, where r is the length parameter on [0, ε). When X is a sphere, then the cone on X is a topological ball. In addition, when X is a standard metric sphere, then the metric cone on X with curvature k is a ball in the space of constant curvature k. Fixing the convention that one-dimensional cone manifolds are circles, we define: Definition. For n 2, a cone n-manifold of curvature k R is a complete metric length space with the following local property: for every point p there exists ε > 0 such that the ball B(p, ε) is isometric to the cone C k,ε (L p ) on some orientable cone (n 1)-manifold L p of constant curvature +1 and diam(l p ) π. We remark that the isometry B(p, ε) = C k,ε (L p ) maps p to the vertex of the cone. Definitions. A cone manifold of curvature k = +1, k = 0 or k = 1 is said to be spherical, Euclidean or hyperbolic, respectively. 11

12 The link of a point p in a cone manifold is the orientable spherical (n 1)-cone manifold L p such that p has a neighborhood isometric to the cone C k,ε (L p ) and p corresponds to its vertex. The link is also called the space of directions, because all minimizing segments starting at the vertex of the cone C k,ε (L p ) = [0, ε] L p / are radius [0, δ] {x} for some x L p. A point is non-singular if it has a neighborhood isometric to a ball in the n- dimensional metric space of curvature k, and it is singular otherwise. The set of singular points is denoted by Σ and it is called singular locus. Remark. A point is non-singular iff its link is a (n 1)-dimensional standard sphere. Example. If M is an n-manifold of constant curvature k and G is a finite group of orientation preserving isometries, then M/G is a cone n-manifold of curvature k (the proof is by induction). Since it is a local notion, an n-orbifold with a metric of constant curvature is also a cone n-manifold. For the proof of the following proposition see for instance [HT]. Proposition 2.1. Any two points p, q of a connected cone manifold of curvature k are joined by a geodesic segment of length pq. In addition if k 0, or if k > 0 and pq < π/ k, then there are a finite number of such geodesic segments. Remark. If σ : [0, l] C is a minimizing segment in a cone manifold C, then all points in σ(0, l) are non singular. This follows easily form the cosinus formula. 2.2 Local description of cone manifolds in dimensions 2 and 3 In dimension 2, the link of a point p is a circle L p of length less than or equal to 2π, because we require diam(l p ) = 1 2 Length(L p) π. The point p is non-singular iff Length(L p ) = 2π. Definition. For a singular point in a 2-dimensional cone manifold, the length of its link is called the cone angle. Remark. In our definition, cone angles are always less than 2π. One could also define cone manifolds with cone angles > 2π, but the metric properties are substantially different (most of the results here are false when cone angles exceed 2π). The justification of the name cone angle comes from the following construction of a cone. We take a sector of angle 0 < α < 2π in the plane of constant curvature k and we identify its faces by a rotation (see Figure 3). In a two dimensional cone manifold, the singular set is discrete. 12

13 Figure 3: In order to understand the local geometry of cone 3-manifolds, we need first to study orientable spherical cone 2-manifolds. We will be mainly interested in cone manifolds with cone angles less than or equal to π. Proposition 2.2. (i) An orientable spherical cone 2-manifold is a 2-sphere with n 0 singular points. (ii) If the cone manifold has cone points with respective cone angles α 1,..., α n, then its area is n 4π (2π α i ) > 0. (iii) The case n = 1 is not possible, and if n = 2 then α 1 = α 2. i=1 Proof. The compactness follows from the fact that a spherical cone manifold is an Alexandrov space of curvature 1 (see Proposition 2.6 below). Since the singular set is discrete, it is finite. Assertions (i) and (ii) follow from Gauss-Bonnet formula. The proof of (iii) will be given in 2.4 below. Corollary 2.3. An orientable spherical cone 2-manifold with cone angles less than or equal to π is one of the following: a standard sphere S 2, a sphere with two cone points S 2 (α, α), a sphere with three cone points S 2 (α, β, γ), with α + β + γ > 2π. Definiton. The cone manifold S 2 (α, β, γ) is called a turnover. It is spherical, Euclidean or hyperbolic according to whether α + β + γ is more than, equal to, or less than 2π, respectively. Corollary 2.4. If p is a point in a cone 3-manifold with cone angles less than or equal to π, then p has a neighborhood isometric to one of the following (see Figure 4): a ball in the space of constant curvature, when L p = S 2 ; a ball with a singular axis, when L p = S 2 (α, α); 13

14 Figure 4: Local description of Corollary 2.4 a ball with three singular edges adjacent to a singular vertex, when L p = S 2 (α, β, γ). Corollary 2.5. The singular locus of a cone 3-manifold with cone angles less than or equal to π is a graph of valency at most three. 2.3 Cone manifolds as Alexandrov spaces A cone manifold with curvature k is an Alexandrov space of curvature k (note that we require the cone angles to be less than 2π). Hence we can use results the following results about Alexandrov spaces (see [BGP]). Proposition 2.6. If C is a cone manifold of curvature k > 0, then diam(c) π/ k. Proposition 2.7. Any triangle in a cone manifold of curvature k > 0 has perimeter 2π/ k. Theorem 2.8. (Angle comparison Theorem) Let be a triangle with geodesic sides γ 1, γ 2, γ 3 and (opposite) angles α 1, α 2, α 3 in a cone manifold of curvature k. If γ 2 and γ 3 are minimizing, and γ 1 π/ k when k > 0, then there exists a comparison triangle in the plane of constant curvature k with the same side lengths, and (opposite) angles α 1, α 2, α 3 satisfying α 2 α 2, α 3 α 3. This theorem is proved in [BGP] in the case where γ 1 is also minimizing. The version stated here follows easily, by dividing γ 1 in a finite number of minimizing segments. A hinge (p; γ, τ) is a figure consisting of two geodesic segments γ and τ withj a common end-point p. The other end-points of γ and τ are denoted by q and r respectively. The following theorem can also be found in [BGP]. Theorem 2.9. (Hinge comparison Theorem) Let (p; γ, τ) be a hinge in a cone manifold of curvature k, such that γ is minimizing and τ π/ k when k > 0. Let ( p; γ, τ) be a hinge in the plane of constant curvature k with the same side lengths and the same angle. If q, r, q, r denote the respective end-points of γ, τ,γ, τ, then pr p r. 14

15 Corollary If C is a spherical cone manifold (of curvature 1) and diam(c) = π, then C is the spherical suspension of a n 1-dimensional spherical cone manifold. Proof. Let p, q C be two points such that d(p, q) = π. Given a third point r C, Proposition 2.7 implies that d(p, q) = d(p, r)+d(q, r). Hence any point in C different from p and q is in the interior of a minimizing segment between p and q. It follows that the cut locus of p is the single point {q} and that C is the spherical suspension on the link of p (or q). 2.4 The cut locus Definition. Given a cone manifold C and a point p C, the cut locus Cut p is the set of points that do not lie in the interior of a segment that minimizes the distance to p. By construction, if p is non singular then Cut p contains the singular locus. When p is non singular, then C Cut p can be embedded in a space of constant curvature k (using Propositions 2.6 and 2.7 when k > 0). When p is singular, then C Cut p can be embedded in a metric cone, (using again Propositions 2.6 and 2.7 when k > 0). Definition. The closure of C Cut p in a space of constant curvature (or in a metric cone) is called the Dirichlet polyhedron centered at p. It can be checked that this is a polyhedron with totally geodesic faces. Now we are ready to prove the last assertion of Proposition 2.2. Proof of Proposition 2.2 (iii). Assume first that C is a cone manifold with two singular points of angles α 1 and α 2. In particular C has area α 1 + α 2. We choose a non-singular point p C and we consider the Dirichlet polygon centered at this point. By Corollary 2.10, we may assume that no point of C is at distance π from p. Let 0 < β 1,..., β l < 2π be the angles of the vertices of the Dirichlet polygon; by Gauss-Bonnet formula: 2π = α 1 + α 2 + l (π β i ). (1) Vertices of the Dirichlet polygon correspond to vertices of Cut p, which can be either smooth or cone points (cone points are always contained in Cut p ). To a smooth vertex of Cut p, there are at least three vertices of the Dirichlet polygon that correspond to it, and the sum of their angles equals 2π. The sum of the angles at the vertices of the Dirichlet polygon associated to a cone point equals the cone angle. Hence equation (1) implies that l = 2 and that β 1 = α 1, β 2 = α 2, because in any other case, the right hand side of the formula would be at least 3π. We have shown that the Dirichlet polygon is a bigon of angles α 1 and α 2. Since it is a bigon in S 2, α 1 = α 2 and its side length is necessarily π. The same argument for an orientable spherical cone manifold with a unique singular point gives a contradiction. 15 i=1

16 From the proof we obtain the following corollary: Corollary (i) A spherical cone 2-manifold with two cone points is obtained by gluing the faces of a bigon with cone angles α. In particular it is the spherical suspension of a circle of length α 2π. (ii) A spherical turnover S 2 (α, β, γ) is obtained by gluing two triangles of angles α/2, β/2, γ/2. In particular spherical cone manifolds with at most 3 cone points are rigid. Proof. Assertion (i) follows from the proof of Proposition 2.2 (iii), and assertion (ii) is proved similarly by considering the Dirichlet polygon centered at a singular point. 2.5 Spherical turnovers As an illustration of comparison theorems we prove the following results Lemma If S 2 (α, β, γ) is a spherical turnover with cone angles α, β, γ π, then diam S 2 (α, β, γ) π/2. Proof. First we bound the distance between two singular points. Let p, q, r be the three singular points of S 2 (α, β, γ). By Corollary 2.11 they form a triangle pqr with angles α 2, β 2, γ 2, which can be isometrically embedded in S2. Since a spherical triangle with angles π 2 has side-lengths π 2, we have that pq, qr, rp π 2. Next we bound the distance xp, when x is non-singular and p is singular. For this we consider the triangle xpq where q is another singular point. We remark that the triangle xpq lies in a triangle pqr, hence we may assume that xpq embeds isometrically in S 2. Since pq π, 2 p π and 2 q π, by spherical trigonometry it follows that 2 xp, xq π. 2 Finally we bound the distance xy between two nonsingular points x and y. We consider the triangle pxy, where p is a singular point. Let p xỹ be the comparison triangle in S 2 (i.e. the triangle with the same side lengths as pxy). By angle comparison (Theorem. 2.8) p p α π. This inequality and p x, pỹ π imply that xy = xỹ π Injectivity radius Definition. The injectivity radius at p is the supremum of all r such that B(p, r) is isometric to the cone C k,r (L p ). The ball B(p, r) = C k,r (L p ) is called standard ball. If we view the tangent space at p as the infinite cone of curvature zero C 0, (L p ), then the injectivity radius can be defined in terms of the exponential map (see [HT] for instance). In particular if p is not singular, then the injectivity radius coincides with the usual injectivity radius. 16

17 For some questions this definition is inconvenient, because there are compact regions with arbitrarily small injectivity radius. For instance, this is the case for nonsingular points close to the singular locus. This motivates the following definition, due to Thurston. Definition. The cone injectivity radius at p is the supremum of all r such that B(p, r) is contained in a standard ball B(q, R) = C k,r (L q ). We will use both notions and we will denote them by classical injectivity radius and cone injectivity radius, in order to be distinguished. Remark. For a singular vertex both notions of injectivity radius coincide. The following lemma follows from elementary trigonometric arguments. Lemma Given ε 1, ε 2 > 0 there exists δ > 0 such that the following holds. Let C be a cone 3-manifold of curvature k [ 1, 1] such that: the cone angles of C are ε 1 at each singular vertex the addition of cone angles is 2π + ε 1. If C contains a point p with cone injectivity radius ε 2, then C also contains a non singular point q with classical injectivity radius δ and such that pq Cone 3-manifolds with boundary Convention. For a cone 3-manifold with totally geodesic boundary, the singular locus will be assumed to be orthogonal to the boundary. To define the cone injectivity radius we use not only standard balls, but also half standard balls with totally geodesic boundary. The double of a half standard ball along its boundary is a standard ball. When it is singular, the boundary of the half standard ball is orthogonal to the singular locus. We remark that the boundary of the cone 3-manifold does not contain singular vertices. Definitions. Let C be a cone 3-manifold with totally geodesic boundary. We define the cone-injectivity radius at a point x C as { } cone-inj(x) = sup δ > 0 B(x, δ) is contained in either a standard ball. or a standard half ball in C Given C a cone 3-manifold with totally geodesic boundary, we define the normal radius of C as the following supremum: { } η( C) = sup r > 0 two segments of length r orthogonal to C which. start at different points of C do not intersect 17

18 3 Compactness and thickness results 3.1 Geometric convergence By geometric convergence we will mean pointed bi-lipschitz convergence. Here we recall some notions. Given ε > 0, a map f : X Y between metric spaces is called (1+ε)-bi-Lipschitz if f is an embedding so that f and f 1 have Lipschitz constant (1 + ε). Hence f is 1-bi-Lipschitz if and only if f is an isometric embedding. A pointed cone manifold is a pair (C, x), where C is a cone manifold and x C is a point. Definition. A sequence of pointed cone 3-manifolds {(C n, x n )} n N converges geometrically to a pointed cone 3-manifold (C, x ) if, for every R > 0 and ε > 0, there exists an integer n 0 such that, for n > n 0, there is a (1 + ε)-bi-lipschitz map f n : B(x, R) C n satisfying: (i) d(f n (x ), x n ) < ε, (ii) B(x n, R ε) f n (B(x, R)), and (iii) f n (B(x, R) Σ ) = f n (B(x, R)) Σ n. Remark. By definition, the following inclusion is also satisfied: f n (B(x, R)) B(x n, R(1 + ε) + ε). 3.2 Compactness Theorem The purpose of this subsection is to establish the analogue of Gromov s compactness theorem for Riemaniann manifolds of pinched sectional curvature (cf. [GLP] and [Pe]) in the context of cone 3-manifolds. Notation. The classical injectivity radius of p is denoted by inj(p), and its cone injectivity radius, by cone-inj(p). Given a > 0, C a denotes the set of pointed cone 3-manifolds (C, p) with constant curvature in [ 1, 1], cone angles in (0, π], and such that p is a smooth point with inj(p) a. Theorem 3.1 (Compactness Theorem). For a > 0, C a is compact for the geometric convergence topology. Proof of Theorem 3.1. The proof requires the following technical lemma: Lemma 3.2. For every R > 0 and a > 0, there exists a constant b > 0 with the following property. For any (C, p) C a and any q B(p, R), cone-inj(q) b. 18

19 A proof of this lemma can be found in [BLP]. Here we give another proof that relies on the results in [BoP, Ch. 3]. Proof of Lemma 3.2. The main point in this proof is to find a uniform lower bound for the classical injectivity radius of a vertex in B(p, R). Once this bound is found, the arguments in [BoP, Ch. 3] apply with minor changes. Since vol(b(p, a)) is bounded below, it suffices to show that there exists a constant C(R) > 0 depending on R such that vol(b(p, R)) inj(q) C(R) for every singular vertex q B(p, R). When q is a singular vertex, inj(q) equals to one of the following: (i) half the length of a geodesic segment with end-points in q; (ii) the length pq of a segment pq such that p Σ is not a vertex, the cone angle at p is π, and pq is orthogonal to Σ at p. (iii) the length pq of a segment pq such that p C and pq is orthogonal to C. In case (i), let σ be a geodesic segment of length σ 1 with end-points in q. By considering the exponential map orthogonal to σ, Lemma 2.12 implies that vol(b(x, R)) is less than the volume of the region bounded by two planes at distance σ intersected with a ball of radius R + 1 centered at its mid-point. Hence vol(b(x, R)) C(R) σ. In cases (ii) and (iii), if σ is a minimizing segment from a singular vertex to a singular edge or circle (or to the boundary), since σ is orthogonal to this edge or circle (or to the boundary), the same argument applies. Using this lemma, the proof of the theorem has three main steps. Given a sequence in C a, first one proves that it has a subsequence that converges to a locally compact metric length space for the Hausdorff-Gromov topology. We remark that cone manifolds with curvature 1 satisfy Gromov pre-compactness criterion, because of Bishop-Gromov inequality on volume of balls (cf. [HT]). The second step consists in showing that the limit of a sequence of pointed cone manifolds in C a is again a cone manifold. This follows from Lemma 3.2, and the fact that the limit of standard balls is again a standard ball. The final step consist in showing that Hausdorff-Gromov convergence in C a implies geometric convergence. This is elementary again by using Lemma 3.2. From Theorem 3.1 and Lemma 2.13, we have the following corollary. This is the result we are going to use in this paper, because the hypothesis about the cone angles are going to be satisfied. Corollary 3.3. (Pre-compactness for cone-injectivity radius) Let (C n, p n ) n N be a sequence of cone 3-manifolds with curvature in [ 1, 1], and possibly with boundary. If there exist constants a > 0, ω > 0 such that: (i) cone-inj(p n ) > a, (ii) the cone angles of C n are ω, and 19

20 (iii) at each singular vertex of C n, the addition of cone angles of the adjacent edges is 2π + ω; then a subsequence of (C n, p n ) n N converges geometrically to a pointed cone 3- manifold. 3.3 Thickness of vertices The following lemma will allow to apply the Compactness Theorem in many situations. Lemma 3.4. (Thick Vertex Lemma) Given 0 < ε < π there is δ = δ(ε) > 0 such that for every cone 3-manifold C with curvature in [ 1, 1], possibly with boundary, the following holds true: if diam(c) 1 and v C is a singular vertex with two adjacent edges of cone angle π ε, then inj(v) δ. Proof. We assume that there is a short segment σ going from a singular vertex v to itself. We take a point p C such that d(p, v) 1 and we consider the triangle with 2 sides pv, σ and vp. By comparison, if σ is short, then the angles between σ and vp are almost π/2 (the shorter is σ, the closer to π/2 are the angles). Note that the angles cannot be bigger that π/2, by Lemma Claim 3.5. Given 0 < ε < π, there is η = η(ε) > 0 such that a spherical turnover S 2 (α, β, γ), with α, β π ε and γ π, has diam S 2 (α, β, γ) π/2 η. Proof of Claim 3.5. It is a case by case argument similar to the proof of Lemma The only extra ingredient is the fact that a spherical triangle with angles α/2, β/2 π/2 ε/2 and γ/2 π/2 has edges of length π/2 η. That follows from elementary trigonometry. Claim 3.5 says that the link of v has diameter less than π η for some constant 2 η > 0 depending only on the constant 0 < ε < π. Hence the angles between σ and vp are at most π η and, by comparison, this implies that the length of σ is bounded 2 below. With a similar argument one can bound the length of an orthogonal segment between v and a singularity, and also between v and C. With these bounds we get the bound for the injectivity radius. More thickness results can be found in [BLP]. 20

21 4 Finiteness properties of cone 3-manifolds 4.1 Euclidean cone 3-manifolds In this paragraph, C denotes a Euclidean cone 3-manifold with cone angles bounded away from 2π (i.e. cone angles α < 2π). In particular C is an Alexandrov space of nonnegative curvature. Similar to the Bonnet-Myers Theorem one can prove: Proposition 4.1. Let C be a Euclidean cone 3-manifold with non-empty 1-dimensional singular set Σ. If the cone angles of C are α < 2π, then: (i) The singular set Σ has finitely many connected components. (ii) The set Σ (0) of singular vertices is finite. Proof of Proposition 4.1. We prove first assertion (i). Seeking a contradiction, we assume that Σ contains infinitely many distinct components {Σ i } i N. We fix a base point p C Σ and for each i N we connect it to Σ i by a shortest segment σ i = px i whose length is denoted by d i and such that x i = Σ i σ i. Let v i be the unit tangent vector to σ i at p pointing towards x i and w i the unit tangent vector to σ i at x i pointing towards p. After passing to a subsequence, we may assume that: (1) v i converges to a unit vector v in the space of directions L p at p: v i v ; (2) d i grows fastly with i to + : d i+1 /d i ր +. We consider the triangle px i x j C, for i < j with j/i large. From (1) we obtain: (3) at the vertex p, the angle p 0. Let p x i x j E 3 be an Euclidean comparison triangle with the same side lengths as the triangle px i x j, then (2) implies: (4) at the vertex x j, the angle xj 0, and therefore p + xi π. Thus angle comparison Theorem 2.8 implies that: (5) p + xi p + xi. Since by (3) p + xi xi and by (4) p + xi π, (5) implies that xi π. This is impossible, because radius(l xi, w i ) max( α, π) < π, where L 2 2 x i is the link of x i and radius(l xi, w i ) is the maximal distance between w i and a point of L xi (the upper bound for the radius uses that w i belongs to the equator of L xi ). The finiteness of the number of vertices follows by an analoguous argument because the diameter of the link of a singular vertex is bounded away from π in terms of α, according to Lemma 4.2 below. Lemma 4.2. Let S be a spherical cone 2-manifold with at least 3 singular points and cone angles α < 2π. Then there is a constant η > 0, depending only on α, such that diam(s) π η. 21

22 The proof of Lemma 4.2 uses comparison Theorems 2.8 and 2.9, and is similar to the proof of Lemma We assume now moreover that Γ is a group acting properly discontinuously on C by isometries. Corollary 4.3. If Γ is infinite and acts cocompactly on C, then C = R Y 2, with Y 2 a closed Euclidean cone 2-manifold, and Γ is virtually cyclic. Proof of Corollary 4.3. After passing to a finite index subgroup one may assume that Γ preserves all components of Σ together with their orientation, and fixes all singular vertices. The proper and discontinuous action of Γ implies that Σ (0) is empty and all components of Σ are lines. Moreover Γ acts by translation along the singular lines, i.e. Γ = Z. If C/Γ is compact (as we assume) then C is quasi-isometric to R and thus has 2 ends. Then the Splitting Theorem applies (cf. [Yam]). 4.2 Hyperbolic cone 3-manifolds with cone angles < π In this section C denotes a hyperbolic cone 3-manifold with cone angles bounded away π and diameter 1. We recall the following definitions: Definitions. A Margulis tube in a hyperbolic cone 3-manifold C is the tubular neighborhood of a totally geodesic cone manifold of small diameter. In our case, a simple closed geodesic (possibly singular) or a hyperbolic turnover transverse to the singularity. A parabolic cusp is a product E 2 [0, + ), where (E 2, g) is a closed Euclidean cone 2-manifold and E 2 [0, + ) has the warped product metric e 2t g dt 2. In our case E 2 is a torus or a Euclidean turnover. Theorem 4.4 (Finiteness Theorem). Let C be a non compact connected hyperbolic cone 3-manifold (without boundary) with cone angles α < π. If C has finite volume, then: (i) the singular locus Σ of C has at most finitely many vertices and components; (ii) C has finitely many ends and all of them are parabolic cusps. The proof of this theorem can be found in [BLP]. Here we need this Finiteness Theorem in 6 only when the following extra hypothesis are satisfied: there exist three constants 0 < ω < α < π and a > 0 such that: the cone angles of C belong to [ω, α], at each singular vertex of C, the addition of cone angles is 2π + ω, and there is a point in C with cone-injectivity radius > a > 0. 22

23 Under these additional hypothesis, there is an alternative proof of Theorem 4.4 following the methods used in [BoP]. For completeness we give this proof. Proof of Theorem 4.4 under the extra hypothesis. We recall that the classical injectivity radius inj(p) at p is the supremum of all r such that B(p, r) is isometric a standard ball. The cone injectivity radius cone-inj(p) at p is the supremum of all r such that B(p, r) is contained in a standard ball. According to these definitions, inj(p) cone-inj(p). For a constant 0 < δ 1, we consider the δ-thick part of C: C [δ,+ ) = {x C cone-inj(x) δ}. The thick part C [δ,+ ) is either compact or empty, because C has a finite volume. The lower bound on cone angles given by the constant ω implies that a standard ball of radius δ has a volume bounded below away from 0. By our assumption, if we choose δ < a, then C [δ,+ ) is not empty. Hence it is compact and the δ-thin part C (0,δ) = {x C inj(x) < δ} has finitely many connected components. Moreover Σ C [δ,+ ) is topologically finite. Since the cone angles are bounded above by α < π, the thick vertex lemma (Lemma 3.4) shows that the cone injectivity radius of every singular vertex is bounded below by a constant depending only on α. We choose δ less than this constant. Then all the singular vertices belong to the δ-thick part of C. Since C [δ,+ ) is compact, there are finitely many singular vertices in Σ. Moreover, one can choose δ < inf{cone-inj(z) z B(y, inj(y)), y Σ (0) }. In this case, any standard ball, centered at a singular vertex y Σ (0) is contained in the δ-thick part of C. Since the δ-thin part C (0,δ) = {x C inj(x) < δ} has finitely many connected components, the proof of the Finiteness Theorem 4.4 follows from their description. This is the content of the following proposition: Proposition 4.5. If δ > 0 is sufficiently small, then a connected component of the δ-thin part C (0,δ) is: or (i) a Margulis tube, with core a simple closed geodesic or a totally geodesic turnover, (ii) a parabolic cusp, with section a torus or a Euclidean turnover. Before starting the proof of Proposition 4.5, we recall the following definition: Definition. If C is a hyperbolic cone manifold with singular set Σ, its developing map and its holonomy are the developing map and holonomy of the non-singular metric on C Σ. Proof of Proposition 4.5. For δ > 0 sufficiently small the δ-thin part is not empty and does not contain any vertex. Since the cone angles are bounded away from 0 and, we can apply the Local Soul Theorem [BoP, Ch. 4] to give a bi-lipschitz approximation of the metric 23

24 structure of a neighborhood of a point x C (0,δ). Then we use these local models to construct foliations of the thin part invariant by the holonomy of C. By the Local Soul Theorem [BoP, Ch. 4], given ε > 0 and D > 1, there is a constant δ 0 = δ 0 (ω, ε, D) > 0 such that: for any 0 < δ < δ 0 every point x C (0,δ) has a neighborhood U x (1 + ε)-bi-lipschitz homeomorphic to the normal cone fiber bundle N ν (S), of some radius ν 1 depending on x, of the soul S of a non-compact orientable Euclidean cone 3-manifold E with cone angles α < π. In addition the (1 + ε)-bi-lipschitz homeomorphism f x :U x N ν (S) satisfies the inequality max ( cone-inj(x), d(f(x), S), diam(s) ) ν/d. Since the cone angles of C are α < π, the possible local models E belong to the following restricted list: (a) S 1 R 2 with (possibly singular) soul S = S 1 {0} ( denotes the metrically twisted product); or a metric product T 2 R with soul a 2-torus T 2 {0}; (b) a metric product (thick turnover) S 2 (α, β, γ) R, with soul a Euclidean turnover S 2 (α, β, γ) {0} (α + β + γ = 2π); (c) the orientable twisted line bundle over the Klein bottle K 2 R. The neighborhood U x is called a (ε, D)-Margulis neighborhood of x. A Margulis neighborhood U x of type (a) is called abelian because U x (Σ U x ) has an abelian fundamental group. Lemma 4.6. The local model of type (c) cannot occur. Proofof Lemma 4.6. Seeking a contradiction, we assume that x C (0,δ) has a neighborhood U x (1+ε)-bi-Lipschitz homeomorphic to a tubular neighborhood N ν (K 2 {0}) K 2 R. Let γ 1 and γ 2 be two simple closed loops in U x, whose images by the (1 + ε)- bi-lipschitz homeomorphism f are simple closed loops in K 2 {0} passing through f(x) and such that f(γ 1 ) preserves the orientation of K 2 while f(γ 2 ) reverses it (but preserves the orientation of C). We can choose γ 1 and γ 2 so that they verify the relation γ 1 γ 2 γ 1 1 = γ 1 2 in π 1 (C Σ, x). Since γ 1 and γ 2 are simple closed loops, they can be homotoped to shortest geodesic loops in U x (which is non-singular), and hence they have non-trivial holonomies. By using the classification of isometries of H 3, the relation above implies that the holonomy of either γ 1 or γ 2 is a rotation of angle π. That contradicts the fact that they can be homotoped to a geodesic loop without intersecting the singular locus Σ. The following lemma is a consequence of [BoP, Prop , Claim 5.5.5]: Lemma 4.7. There is a choice of the constants ε > 0 and D > 1 in the local soul theorem (depending only on ω) so that if a point of a connected component N of C (0,δ) has a (ε, D)-Margulis neighborhood of type (b), then every point in N has a 24

25 (ε, D)-Margulis neighborhood of type (b). Moreover N is homeomorphic to a product S 2 (α, β, γ) (0, + ). According to Lemmas 4.6 and 4.7, the proof of Proposition 4.5 splits into two cases: Case a) Every point of the connected component N of C (0,δ) has an abelian Margulis neighborhood. Then in this case N is either a Margulis tube with soul a (possibly singular) closed geodesic or a parabolic cusp with horospherical section a 2-torus. Case b) Every point in N has a Margulis neighborhood of thick turnover type (b). Then in this case N is either a Margulis tube with soul a totally geodesic turnover or a parabolic cusp with horospherical section a Euclidean turnover. 4.3 Proof of Proposition 4.5 in case a) Let N be a connected component of the thin part C (0,δ). We shall construct a foliation of N by tori or circles, by using the abelianity of the holonomy of π 1 (U x Σ U x ) for a Margulis nieghborood U x of every x N. We remark that the holonomy of π 1 (U x Σ U x ) is non trivial, because some points in U x Σ U x have a shortest non-singular loop. The key lemma for the construction of the foliation on N is: Lemma 4.8. For every x N, the holonomy of π 1 (U x Σ U x ) is contained in a complex 1-parameter subgroup G x of PSL 2 (C). This subgroup G x is the identity component of the commutator in PSL 2 (C) of the holonomy of any element in π 1 (U x Σ U x ) that has a non-trivial holonomy. In particular either: (i) G x = C is the group of isometries that preserve an oriented geodesic in H 3, or (ii) G x = C is a group of parabolic isometries fixing a point in H 3. Proof of Lemma 4.8. Since π 1 (U x Σ U x ) has non-trivial abelian holonomy, the classification of the abelian subgroups of PSL 2 (C) shows that the holonomy of π 1 (U x Σ U x ) is contained in one of the folowing groups: (i) a group of elliptic or hyperbolic transformations fixing two points of H 3 ; (ii) a group of parabolic transformations fixing a point in H 3 ; (iii) a finite group Z/2Z Z/2Z of rotations of angle π along three geodesic that intersect orthogonally at one point. To prove Lemma 4.8 we just have to observe that case (iii) cannot occur, since some points of U x Σ U x have a shortest non-singular loop contained in U x Σ U x. Such a geodesic loop cannot have a rotation of angle π as holonomy. Let N be the closure of N. According to our definitions of injectivity radius inj(x) and cone-injectivity radius cone-inj(x) at a point x N, inj(x) cone-inj(x). Moreover one has: 25