Ortholengths and Hyperbolic Dehn Surgery

Transcription

1 Ortholengths and Hyperbolic Dehn Surgery James Geoffrey Dowty July 2000 Department of Mathematics and Statistics The University of Melbourne Submitted in total fulfilment of the requirements of the Degree of Doctor of Philosophy

2 Dedicated to my parents Barry and Lisa Dowty for their love and encouragement 1

3 Abstract This thesis studies the hyperbolic Dehn surgery space H(M) of incomplete, finite-volume hyperbolic structures on an orientable 3-manifold M which admits a complete 1-cusped hyperbolic structure. We define an ortholength invariant on H(M) which takes values in a complex affine algebraic variety P. This invariant is easily computable and it locally parameterises H(M). For (topological) Dehn fillings of M, the ortholength invariant is a complete invariant and it determines the tube radii about the core geodesics of all but a finite number of Dehn fillings. We give a closed expression for these tube radii in terms of this invariant. Underlying this work is a geometrical construction based on the possiblysingular core which arises from generalised hyperbolic Dehn surgery on M. Given a hyperbolic structure there is a natural spine for M whose complement is the interior of a tube domain D analogous to a Dirichlet domain based on the singular core. This domain D is a hyperbolic solid torus and the hyperbolic structure on M can be recovered from D by identifying points of its boundary. Conversely, we can try to reconstruct a hyperbolic structure on M from its ortholength invariant. To each point of P we give a representation π 1 (M) PSL 2 (C). We then describe a method for constructing a corresponding hyperbolic structure on M via its tube domain. This construction is quite general and it succeeds in many cases where traditional methods fail. 2

4 This is to certify that 1. this thesis comprises only my original work except where indicated in the preface, 2. due acknowledgement has been made in the text to all other material used, 3. this thesis is less than 100,000 words in length, exclusive of tables, maps, bibliographies and appendices. James Dowty 3

5 Preface The major original contributions of this thesis are contained in Sections 1.3, 3.4, 3.5, 4.3, 4.4, 4.6, 5.5, 5.6 and 6.3. Sections 1.1, 1.2, 2.1, 4.2 and 5.2 are largely or entirely reviews of known results (as indicated in the text). The material presented in Sections 6.1 and 6.2 is based on collaborative work with Oliver Goodman and Craig Hodgson. 4

6 Acknowledgements Most importantly, I would like to thank my supervisor, Craig Hodgson. A doctorate is in many ways a research apprenticeship and Craig has always been a patient and constructive teacher. Without his guidance and enthusiasm I would have lost my way many times and run out of energy many years ago. I am also very grateful to my parents for nurturing my interest in the world and for always treating me with understanding and respect. They and the rest of my family have constantly supported me in ways which range from empathy and encouragement to practical help with the day-to-day matters of life. I would also like to thank my friends for keeping me on my toes! I m very lucky to know so many good people who broaden my horizons, don t let me take myself too seriously and keep me excited about life. This work was partially supported by an Australian Postgraduate Award. 5

7 Contents Introduction 9 1 Configurations of Lines in Hyperbolic 3-Space Dimensional Hyperbolic Geometry Line Matrices Hextets and Right-Angled Hexagons Surfaces Equidistant from Two Lines Two Models of Hyperbolic 3-Space Explicit Equations for Equidistant Surfaces Intersections of Equidistant Surfaces Tube Domains Simple Closed Geodesics in Closed Hyperbolic 3-Manifolds Geodesics in a Singular Hyperbolic 3-Manifold The Tube Domain of a Singular Hyperbolic 3-Manifold Properties of Tube Domains The Canonical Dual Ideal Triangulation Parameterising Hyperbolic Structures with Ortholengths Ortholengths of Smooth Structures The Deformation Space of Hyperbolic Structures on M The Ortholength Invariant Ortholengths and the Character Variety Continuity of Tube Domains Tube Radii

8 5 The Figure-8 Knot Complement The Ortholength Space of the Canonical Triangulation The Character Variety The Presentation of π 1 (M, ) based on K The Natural Isomorphism between the Two Presentations The Ortholength Invariant Tube Radii Constructing Hyperbolic Structures via Tube Domains Constructing Hyperbolic Tubes Face-Pairings Sufficient Conditions for Obtaining Hyperbolic Structures Directions for Future Research A Trace Calculations for the Figure-8 Knot Complement 144 References 148 7

9 List of Figures 1.1 The two orientation-preserving classes of orthogonal frames Arrangements of (a) three lines and (b) four lines The extremities of l ij A neighbourhood of the ball B in H The set-up for Lemma The two types of neighbourhoods of singular points in a standard spine The portion of Q Σ contained in one of the tetrahedra of the dual ideal triangulation of M The figure-8 knot The canonical ideal triangulation K for the figure-8 knot complement The Wirtinger generators for π 1 (M, ) The tetrahedra of K, truncated to show the edges of the corresponding cell-decomposition L of N The cell-decomposition L of N, viewed from outside of M The maximal tree of L and the generators for π 1 ( N, ) The edge classes and the maximal tree in S The loops corresponding to the generators of the presentation for π 1 (M, ) based on K Tube domains for cone-manifolds singular along the figure-8 knot complement Tube domains for m007(3,1) and some related cone-manifolds The notation used in the proof of Theorem

10 Introduction Let M be an orientable 3-manifold which admits a complete, finite-volume hyperbolic structure with a single cusp. Thurston s hyperbolic Dehn surgery theorem applied to M says that all but a finite number of topological Dehn fillings on M have hyperbolic structures. To prove this, Thurston introduced the deformation space H(M) of incomplete hyperbolic structures on M whose completions have special kinds of singularities (see [44]). This space is of continuing interest because it offers a possible approach (due to Thurston) to the Geometrization Conjecture. In particular, non-hyperbolic geometric structures and topological decompositions along incompressible spheres and tori can often be produced by understanding the ways that hyperbolic structures can degenerate near the boundary of H(M) (see Thurston [44, Ch. 4], Kerckhoff [24] and Kojima [28]). This approach has recently yielded a proof of the Orbifold Theorem (see Cooper-Hodgson-Kerckhoff [9]). A hyperbolic structure in H(M) induces a metric on M and we let M t denote the corresponding (incomplete) metric space and let M t = M t Σ t denote its completion. Such spaces M t have a hyperbolic structure which is smooth everywhere except possibly along the distinguished set Σ t, where Dehn surgery-type singularities may occur. Locally the geometry along Σ t is modelled on one of the following: 1 a neighbourhood of a simple closed geodesic in a hyperbolic 3-manifold, 2 a hyperbolic cone-manifold with singular set a simple closed geodesic, 3 a cone over a torus with a hyperbolic metric singular at the cone point. In Chapter 3 we define a canonical fundamental domain D t for (M t, Σ t ) called the tube domain of M t. The interior of this domain can be identified with 9

11 the set of points in M t which possess a unique distance-realising geodesic from themselves to Σ t. Hence D t is a kind of Dirichlet domain based on Σ t. It is closely analogous to the Ford domain of the complete hyperbolic structure on M, whose interior can be identified with the set of points of M which possess a unique distance-realising geodesic from themselves to a horoball neighbourhood of the cusp. In fact, in Section 4.5 we will show that the tube domains vary continuously with the hyperbolic structure on M and that the tube domains approach the Ford domain as the structure becomes complete. If the hyperbolic structure of M t extends smoothly to M t then we can give a more concrete description of D t. In this case there is a covering isometry π : H 3 M t and π 1 (Σ t ) is a set of disjoint lines in H 3. Then D t is (the closure of) the set of points of H 3 which are closer to a line Σ π 1 (Σ t ) than to any other line of π 1 (Σ t ), modulo the deck-transformations which preserve Σ (see Section 3.1). The boundary of D t has a natural decomposition into faces and M t is recovered from D t by isometrically identifying its faces in pairs. This gives a surjection D t M t which takes the boundary of D t to a naturally occurring spine in M t. Generically there is a canonical ideal triangulation for M dual to this spine, so this give rise to an algorithm to find an ideal triangulation for the complement of a simple closed geodesic in a closed hyperbolic 3-manifold. This algorithm has been implemented as part of a computer program called tube by Goodman-Hodgson [15] and it has been used by Miller [32] to study the topological properties of simple closed geodesics and their complements. Meyerhoff has defined the complex ortholength spectrum of a closed hyperbolic 3-manifold as the set of complex distances between pairs of the manifold s simple closed geodesics (see [31]). He has shown that the ortholength spectrum plus some combinatorial data determines the manifold up to isometry. In Chapter 4 we define an invariant Orth K : H(M) C n called the ortholength invariant (see Section 4.3). This invariant is defined in terms of an ideal triangulation K of M, but it is actually independent of K in the sense that for sufficiently general K, Orth K determines the ortholength invariant corresponding to any other ideal triangulation (see Corollary 4.8). For a hyperbolic structure whose metric completion is a cone-manifold, the ortholength invariant is essentially a list of hyperbolic cosines of the complex 10

12 distances 1 from the cone-manifold s singular set to itself along the edges of K. We show that the ortholength invariant is an algebraic map which locally parameterises H(M). We also show that Orth K is a complete invariant of the hyperbolic structures on M which correspond to topological Dehn fillings (see Proposition 4.6). This implies that a finite (and computable) subset of the ortholength spectrum, plus slightly stronger combinatorial information than assumed by Meyerhoff in [31], determines the filled manifold up to isometry. There is a close connection between the ortholength invariant of M t and the tube radius of M t, i.e. the radius of the largest embedded tube about Σ t in M t. As a corollary to the fact that the tube domains vary continuously, we show that if the combinatorics of the Ford domain are generic (i.e. if Epstein-Penner s cell-decomposition [11] is an ideal triangulation) then the ortholength invariant determines the tube radii of all hyperbolic structures in a neighbourhood of the complete structure. For these structures, we give an exact expression for the tube radius in terms of the traces of certain holonomies (see Proposition 4.4 and Theorem 4.13, cf. Section 5.6). The importance of the tube radius has emerged recently from the work of Kojima [27] and Hodgson-Kerckhoff [24], [23]. In these papers, the condition that the tube radius stays bounded away from zero as cone-angles are varied has emerged as the key to ensuring that the hyperbolic structures do not degenerate. (By contrast, it is possible for hyperbolic cone-manifolds to degenerate while their volume remains bounded above zero.) This suggests a relationship between the boundary of H(M) and the ortholength invariant which deserves further study. We also consider the converse problem of constructing incomplete hyperbolic structures on M via their tube domains. The main technique for constructing hyperbolic structures with cone-type or Dehn surgery-type singularities has been based on an ideal tetrahedral decomposition of the manifold M. The idea for this approach came from Thurston and it has been automated by Weeks in his program SnapPea [48]. SnapPea tries to find hyperbolic structures on M in terms of complex tetrahedral parameters which satisfy certain consistency equations. Whenever the solutions to these consistency equations are tetrahedral parameters with positive imaginary parts, they correspond to a genuine decomposition of M into geometrical parts. How- 1 The complex distance between two lines is the hyperbolic distance between them plus i times an angle of twist see Definition

13 ever, when the consistency equations have solutions with negative imaginary parts then there is no obvious interpretation of the decomposition of M and Thurston s method fails to give a corresponding hyperbolic structure. While SnapPea is very robust and can calculate many important invariants (e.g. volume, the Chern-Simons invariant, the length spectrum) it is known that SnapPea sometimes fails to find hyperbolic structures (see the introduction of Chapter 6 for examples) and it also does not calculate the tube radius. Other techniques for constructing hyperbolic structures with cone-type singularities have been in terms of Dirichlet domains (e.g. see [21] and Suárez-Peiró [37]). Suárez-Peiró [37] gives a general algorithm for calculating hyperbolic structures with given cone-angles smaller than 2π, and her methods are especially suited to analysing the degeneration of hyperbolic cone-manifolds to other geometrical structures. In Chapter 6 we describe a new method for calculating hyperbolic structures with given Dehn surgery-type singularities on a manifold M (as above, we only consider the case when M has a single end). This method attempts to find hyperbolic structures on M by constructing their corresponding tube domains and face-pairings. This approach has been automated by Goodman- Hodgson in a computer program [15] called tube. This program has been used to calculate hyperbolic structures which SnapPea fails to compute (see the examples of Chapter 6) and it naturally lends itself to calculating the tube radius. This thesis is set out as follows. In Chapter 1 we show that it is possible to parameterise a configuration of lines in H 3 in terms of the complex distances between them, which is the key to proving that the ortholength invariant locally parameterises H(M). The surfaces which occur in the boundaries of the tube domains are the equidistant surfaces between two lines in H 3, and in Chapter 2 we give a detailed description of such surfaces and their intersections. In Chapter 3 we prove some elementary properties of tube domains and show that they give rise to dual ideal triangulations. The main chapter of this thesis is Chapter 4, where we develop the theory of the ortholength invariant, prove that tube domains approach the Ford domain, and give a formula for the tube radius. Chapter 5 is devoted to studying the results of Chapter 4 in the case where we take M to be the figure-8 knot complement. In particular we calculate the ortholength invariant and the tube radius (in a neighbourhood of the complete case). In Chapter 6 we 12

14 give a description of the theoretical underpinnings of the program tube before finishing with Section 6.4 which discusses some questions raised by the work of this thesis. 13

15 Chapter 1 Configurations of Lines in Hyperbolic 3-Space We say that n lines l 1,..., l n in hyperbolic 3-space realise a set of complex numbers x ij C (i, j = 1,..., n) if cosh of the complex distance 1 between each pair of lines l i and l j is equal to x ij. This chapter is motivated by the following question. Problem 1.1 When is it possible to realise a given set of complex numbers x ij C (i, j = 1,..., n) by an arrangement of lines and how unique is such an arrangement? A complete answer to Problem 1.1 is given in Theorem 1.9. A corollary to this theorem says that there exists an arrangement of four lines which realises a set of complex numbers x ij C (i, j = 1,..., 4) if and only if these complex numbers satisfy a hextet equation (see Proposition 1.11). This corollary is the key result which is used in Chapter 4 to construct a birational equivalence between certain irreducible components of the PSL 2 C-character variety of a 1-cusped hyperbolic 3-manifold M and a certain algebraic variety P 0 (K) 1 The complex distance between two lines is the hyperbolic distance between them plus i times an angle of twist see Definition

16 defined in terms of hextet equations. This corollary will also be used (in Chapter 6) to try to construct incomplete hyperbolic structures on M from the points of P 0 (K). This chapter begins with a brief review of hyperbolic geometry (Section 1.1) before setting up some algebraic tools (see Section 1.2) which are needed to solve Problem 1.1 (see Section 1.3) Dimensional Hyperbolic Geometry This section gives a quick exposition of the aspects of 3-dimensional hyperbolic geometry which are used in this chapter. For proofs of the facts quoted below, see Benedetti-Petronio [2], Ratcliffe [40] or Thurston [46]. Throughout this chapter, H 3 will denote the upper half-space model for 3- dimensional hyperbolic space. This is the set {(z, t) C R t > 0} endowed with a Riemannian metric of constant negative sectional curvature. There is a natural boundary S 2 of H 3 called the sphere at infinity which consists of the complex plane C {0} plus an extra point which is needed to compactify {(z, t) C R t 0}. The decomposition S 2 = C allows us to identify S 2 with the complex projective line CP 1. This identification is natural because each orientation preserving isometry of H 3 extends to S 2 = CP 1 as a unique projective transformation. Conversely, each projective transformation of CP 1 acts on S 2 and this action extends uniquely to an orientation preserving isometry of H 3. Hence the orientation preserving isometries of H 3 can be identified with the group of projective transformations of CP 1. This reduces the study of 3- dimensional hyperbolic geometry (in the sense of Klein s Erlanger program for geometry see [25] and [20]) to the study of the properties of CP 1 which are preserved by the group of projective transformations. In practice, this view of 3-dimensional hyperbolic geometry is especially useful when considering lines or convex ideal polyhedra in H 3 because these objects can be identified with finite subsets of S 2. In the context of 3-dimensional hyperbolic geometry it is customary to refer to the projective transformations of CP 1 as the Möbius transformations. 15

17 Recall that a Möbius transformation is given by the following action of a matrix [ ] a b c d GL2 C on CP 1 (where z C CP 1 ): z = (z : 1) (az + b : cz + d) = az + b C if cz + d 0, cz + d and = (1 : 0) (a : c) = a/c C if c 0. A (non-zero) multiple of [ ] a b c d has the same action, so we identify the Möbius transformations (and hence the orientation preserving isometries of H 3 ) with PSL 2 C, the quotient of SL 2 C by the subgroup {±1}. In the later sections of this chapter we will also assume the following elementary facts of 3-dimensional hyperbolic geometry: There is a unique isometry which takes any three points of S 2 to three other points. Two distinct points of S 2 specify a line in H 3 and conversely any line determines two distinct points of S 2 (the end-points of the line). Any two lines in H 3 with distinct end-points have a unique commonperpendicular. H 3 is homogeneous and isotropic. 1.2 Line Matrices In this section we define a correspondence between oriented lines in H 3 and the traceless elements of SL 2 C. We then define the complex distance between two oriented lines and relate this notion to a certain bilinear form on the vector space of 2 2 traceless complex matrices. Other than Lemma 1.6, these results are simply re-statements of some of the ideas Fenchel presented in [12]. For related material see Thurston [46, ]. We first define an orientation on a line in H 3 as a choice of ordering of its endpoints. Since the end-points of a line in H 3 determine the line, we can specify 16

18 any oriented line l by giving its ordered end-points (p, q) S 2 S 2 \, where = {(x, x) x S 2 }. Now, a simple calculation shows that every traceless matrix A SL 2 C has eigenvalues ±i. The eigenvectors of A correspond to points on S 2 = CP 1 which are fixed by the action of A. We define the oriented line corresponding to A to be the line with ordered end-points (p, q) S 2 S 2 \, where p corresponds to eigenvalue i and q corresponds to eigenvalue i. Conversely, to each oriented line there is a unique traceless matrix in SL 2 C. For if (p, q) S 2 S 2 \ are the ordered end-points of an oriented line then consider the matrix [ ] i 0 g g 1 (1.1) 0 i where g PSL 2 C is any isometry which takes 0, to p, q (respectively) and g SL 2 C is either of the matrices which covers g. Note that the action of the traceless matrix [ ] i 0 0 i SL2 C on S 2 fixes 0, S 2, with 0 corresponding to eigenvalue i and corresponding to eigenvalue i. Hence the matrix (1.1) belongs to SL 2 C, is traceless, and its action fixes p and q, with p corresponding to eigenvalue i and q corresponding to eigenvalue i. To see that this matrix is unique, note first that we can assume without loss of generality that (p, q) = (0, ). Any matrix [ ] a b c d GL2 C which fixes 0 and must be diagonal, i.e. b = c = 0. The trace of this matrix is therefore a+d and its determinant is ad. Hence there are only two traceless matrices in SL 2 C [ which fix 0 and. Explicitly considering these two matrices then shows that i 0 ] 0 i is the only one whose i eigenvector corresponds to 0 S 2. Hence we have identified the set of oriented lines in H 3 with SL def = {l SL 2 C tr l = 0}, the traceless elements of SL 2 C. Note from the description (1.1) that each matrix l SL acts on H 3 as a half-turn about the line corresponding to l. From now on we will use the same symbol to denote an oriented line and the corresponding element of SL. Note that as a consequence of this, l SL denotes the same line in H 3 as l SL but with the opposite orientation. Now, PSL 2 C acts by isometries on the set of oriented lines in H 3. Under the correspondence between oriented lines and SL, this gives us a corresponding 17

19 action of g PSL 2 C on SL given by g l = gl g 1 for any l SL, where g SL 2 C is either of the matrices covering g. Hence studying the geometric properties of arrangements of lines in H 3 is the same as studying properties of subsets of SL which are invariant under the conjugacy action of SL 2 C. Given two oriented lines l, m S, an obvious conjugacy invariant of the pair is tr(lm). We will see that this invariant is essentially the hyperbolic cosine of the complex distance between l and m, which we now define. Let l, m SL be two oriented lines which have no end-points in common. Then l and m have a unique unoriented common-perpendicular n. There is an isometry g PSL 2 C taking the end-points of n to 0 and (in some order) and taking the second end-point of l to 1. Then g is unique up to possibly multiplying it on the left by an isometry which swaps 0 and and keeps 1 fixed (i.e. up to the isometry which acts as z 1/z on S 2 ). Now, all lines perpendicular to the line with end-points 0 and are invariant (though orientations are reversed) under the half-turn which fixes 0 and. This half-turn acts as z z on S 2 so the ordered end-points of g l must be ( 1, 1). Similarly, the ordered end-points of g m are ( α, α) for some α C \ {0}. The only indeterminacy in α comes from the non-uniqueness of g (as described above). Hence there is a well-defined association from l and m to α ±1. Definition 1.2 (Complex Distance) Let l, m SL be two oriented lines which have no end-points in common. Then with α ±1 C \ {0} as defined in the above two paragraphs, define the complex distance between l and m to be d C (l, m) = ± log α. Note that while d C (l, m) is only defined up to multiplication by ±1 and the addition of an element of 2πiZ, cosh(d C (l, m)) is a well-defined complex number. If l, m SL share one or both end-points 2 then we define d C (l, m) to be 0 if both lines are both oriented towards or away from their common end-point, 2 The definition of d C (l, m) in the case where l and m share one or both end-points is motivated by Proposition 1.4 (below). 18

20 and to be iπ if one is oriented towards and the other away from their common end-point. It is not hard to prove that cosh(d C (l, m)) = cosh(d C (m, l)). We now define our main algebraic tool for addressing Problem 1.1. Definition 1.3 (Line Matrices) Define the vector space L of line matrices to be all 2 2 complex matrices with zero trace. This space is endowed with a symmetric bilinear form <, > given by <l, m> = 1 2 tr(lm) for any l, m L. A bilinear from g on a vector space V defines a map from V to the dual V of V given by v g(v, ). We say that g is non-degenerate if this map is an isomorphism. Consider the isomorphism L C 3 defined by [ ] a b c a a b c (which we denote by l [l]). Then <l, m> = [l] T /2 [m] 0 1/2 0 for any l, m L. From this it is clear that <, > is non-degenerate. Note that each non-singular line matrix l L acts on H 3 as a half-turn about some line since a scalar multiple of l has the same action as l itself and ±l/ det l SL. Hence there is an (unoriented) line associated with each non-singular line matrix. This is where the terminology line matrices comes from. 19

21 A simple calculation shows that <l, l> = det l so SL = {l L <l, l> = 1} is the set of normalised line matrices and l L is a singular matrix if and only if <l, l> = 0. Now, the action of PSL 2 C on SL extends to L in an obvious way 3 and clearly <, > is invariant under this action. Also note that <, > is invariant under the map L L given by l l. Lemma 1.6 (below) essentially says that these are the only isomorphisms of (L, <, >). Before proving this lemma, however, we will prove a fundamental link between (L, <, >) and the geometry of lines in H 3, namely Proposition 1.4. Proposition 1.4 For any l, m SL, <l, m> = cosh(d C (l, m)). (1.2) Note that by the bilinearity of <, >, Proposition 1.4 implies that for any l, m L, <l, m> = l m cosh d where l and m are choices of square root of <l, l> and <m, m> (respectively) and d = d C (l/ l, m/ m ). Proof of Proposition 1.4: To begin with, assume l, m SL have no end-points in common and so have an (unoriented) common-perpendicular, n. There is an isometry taking the end-points of n to 0, (in either order) and taking the second end-point of l to 1. As argued in the lead-up to Definition 1.2, this isometry therefore takes l and m to the lines with ordered end-points ( 1, 1) and ( α, α) respectively (for some α 0). Since the left- and right-hand sides of <l, m> = cosh(d C (l, m)) are invariant under the action of PSL 2 C, it s enough to prove (1.2) in this special case. Now, for some choice of sign for d C (l, m) (see Definition 1.2), α = e d C(l,m) 3 For g PSL 2 C and l L, g l = gl g 1 where g SL 2 C is one of the two matrices covering g. 20

22 and an easy calculation shows that [ ] 0 i l = and m = i 0 [ ] 0 iα. i/α 0 Hence <l, m> = 1 2 tr(lm) = 1 2 (α + 1/α) = cosh(d C(l, m)) as required. If l, m SL share one end-point and both lines are oriented towards this shared end-point then up to isometry we can assume l is the oriented line with ordered end-points (0, ) and that m has ordered end-points (1, ). Then l = [ ] [ i 0 i 2i ] 0 i and m = 0 i so 1 tr(lm) = tr[ ] = 1 = cosh(dc (l, m)) since d C (l, m) = 0, so (1.2) holds in this case. Setting l = [ ] i 0 0 i and carrying out a similar calculation proves (1.2) in the case when l and m share an end-point but one is oriented towards and the other is oriented away from it. If l and m share both end-points then l = ±m so <l, m> = ±1 depending on whether or not the orientations of l and m agree or disagree. Hence (1.2) holds in this case as well. We have the following corollary to Proposition 1.4. Corollary 1.5 Let l, m L be two non-singular line matrices. Then <l, m> = 0 l and m correspond to perpendicular lines in H 3. It is now simple to prove Lemma 1.6 For each linear map φ : L L which preserves <, > there is some element g PSL 2 C and some choice ±1 of sign so that for any l L. ±φ(l) = g l Proof: Consider three lines l 1, l 2, l 3 SL for which <l i, l j > = 0 for each i j. By Corollary 1.5, these lines must all meet at a common point of H 3 where they are perpendicular. We assume that the orientations of the l i have been chosen so that they define a right-handed frame at this common point (see Figure 1.1). 21

23 l 3 l 1 l 2 l 1 l 2 l 3 (a) Left-handed (b) Right-handed Figure 1.1: The two orientation-preserving classes of orthogonal frames Now, if φ : L L preserves <, > then <φ(l i ), φ(l j )> = 0 for each i j and so φ(l 1 ), φ(l 2 ), φ(l 3 ) are also mutually perpendicular lines which meet at a common point. By replacing φ by φ if need be we can assume φ(l 1 ), φ(l 2 ), φ(l 3 ) define a right-handed frame. Since PSL 2 C acts transitively on the bundle of right-handed orthogonal frames of H 3 (see [46, 2.2]) there is an isometry g PSL 2 C which takes l i to ±φ(l i ) for each i = 1, 2, 3. Hence ±φ(l i ) = g l i for each i = 1, 2, 3. Since the l i form a basis for L (linear dependence would imply that <l i, l i > = 0 for some i) this proves the lemma. In fact, from this proof it is clear that the group PSL 2 C {±1} acts faithfully on L and so this group may be identified with the isomorphisms of (L, <, >). Note that 1 acts by simultaneously reversing the orientations of all lines and is not related to the orientation-reversing isometries of H 3. From Corollary 1.5 we can see why generically there is a unique unoriented common-perpendicular to two lines l, m SL for which l ±m. Since L is 3- dimensional and <, > is non-degenerate, the kernels of the linear functionals <l, > and <m, > meet in a 1-dimensional subspace, spanned by n L, say. Hence there can be at most one (unoriented) common-perpendicular between l and m. If n is non-singular then n corresponds to an unoriented line in H 3 which is perpendicular to l and m. If n is singular then there is no commonperpendicular to l and m. A more careful analysis, using the fact that we 22

24 can take n to be lm ml, shows that n is singular if and only if l and m share an end-point. To finish this section, we now use Proposition 1.4 to rephrase Problem 1.1 as follows. Problem 1.7 Given some complex numbers x ij C (i, j = 1,..., n) so that x ij = x ji and x ii = 1, do there exist oriented lines l 1,..., l n SL so that <l i, l j > = x ij? To what extent do these conditions determine the lines l 1,..., l n if they exist? 1.3 Hextets and Right-Angled Hexagons Via the correspondence between (L, <, >) and the geometry of lines in H 3 which we developed in the last section (especially Proposition 1.4 and Lemma 1.6) we can now use a simple result from linear algebra (Lemma 1.8) to neatly solve Problem 1.7. Let (V, g) be a complex vector space of (finite) dimension n equipped with a symmetric, bilinear form g. Then associated to g there is a map V V from V to its dual given by v g(v, ). We define the rank of g (denoted rank(g)) to be the dimension of the image of this map. A more easily computable definition of rank(g) can be given in terms of an isomorphism ψ : V C n which takes a vector v V to a column matrix [v] ψ C n. We say that an n n matrix G represents g with respect to an isomorphism ψ : V C n if g(u, v) = [u] T ψ G[v] ψ for every u, v V. Then rank(g) is the rank of any matrix G which represents g (with respect to some isomorphism V C n ). Recall the following terminology: g is non-degenerate if its associated map V V is an isomorphism, i.e. of rank(g) = n; an isomorphism of the pair (V, g) is any linear map V V which preserves g; a vector v is null if g(v, v) = 0; and the orthogonal complement V0 of a subspace V 0 V is def = {v V g(u, v) = 0 for all u V 0 }. V 0 23

25 Lemma 1.8 Let (V, g) and (W, h) be two complex vector spaces equipped with symmetric, bilinear forms g and h. If rank(g) rank(h) then there exists a linear transformation φ : V W so that φ h = g i.e. for any x, y V, g(x, y) = h(φx, φy). Further, if h is non-degenerate and rank(g) = rank(h) or rank(g) = rank(h) 1 then φ is unique up to composing it on the left with an isomorphism of (W, h). Proof: First note that if n = dim(v ) then there is an isomorphism ψ g : V C n so that g is represented (with respect to ψ g ) by the diagonal matrix diag(1,..., 1, 0,..., 0) with r = rank(g) non-zero entries. The existence of such an isomorphism is equivalent to the existence of an orthogonal basis for V which has exactly r non-null vectors. To prove the existence of such a basis, first set V 0 = {0} and choose some non-null vector v 1 V0 (of course V0 = V ). Now, if all vectors in a subspace of V are null then g(u, v) = 0 for any pair of vectors in that subspace (expand out g(u+v, u+v)). So if such a v 1 doesn t exist then g can be represented by the diagonal matrix diag(0,..., 0). Now, let V 1 be the span of v 1 and choose some non-null vector v 2 V1. If such a v 2 doesn t exist then g can be represented by diag(1, 0,..., 0). If v 2 does exist then let V 2 be the span of v 1 and v 2 and then continue as above. At the k th iteration of this procedure we will have a subspace V k spanned by k non-null, mutually orthogonal vectors v 1,..., v k. If all the vectors of Vk are null then g can be represented by a matrix diag(1,..., 1, 0,..., 0) which has k non-zero entries. If there is some non-null vector v k+1 V k define V k+1 to be the span of v 1,..., v k+1. Now Vk but v k+1 Vk+1, so dim(vk+1 ) = dim(v k then Vk+1 since v k+1 Vk ) 1. Hence this procedure must terminate after at most n steps, and at this stage we will have shown that g can be represented by some diagonal matrix diag(1,..., 1, 0,..., 0). It is clear that this diagonal matrix must have exactly r = rank(g) non-zero entries. Similarly, if m = dim(w ) then there is an isomorphism ψ h : W C m so that h is represented (with respect to ψ h ) by the diagonal matrix diag(1,..., 1, 0,..., 0) which has rank(h) non-zero entries. 24

26 Now, let e 1,..., e p denote the standard basis for C p (any p > 0) and assume rank(g) rank(h). We define φ : C n C m to be the map which takes e 1,..., e r C n to e 1,..., e r C m (respectively) and takes e r+1,..., e n C n to 0 C m. Then clearly φ def = ψ 1 h φ ψ g satisfies φ h = g. Now, if φ : V W is any map for which (φ ) def h = g then φ = ψ h φ ψg 1 is a map C n C m which preserves the bilinear forms given above 4. Hence the vectors φ e 1,..., φ e r form an orthonormal set in C m and for j = r +1,..., n, each φ e j is a null-vector. Now, suppose that rank(h) = m and that r = m or r = m 1 (where r = rank(g), as above). Then we first note that φ e j = 0 for each j = r +1,..., n. For let v i denote φ e i for each i = 1,..., r. If r = m then v 1,..., v m form an orthonormal basis for C m. If r = m 1 then there is some vector v m C m so that again v 1,..., v m form an orthonormal basis. (Such a v m exists because the orthogonal complement of the span of v 1,..., v m 1 is one-dimensional 5 and since rank(h) = m it cannot consist entirely of null-vectors.) Now, for any j = r+1,..., n let u = φ e j. Expressing u in terms of the basis v 1,..., v m gives u = u 1 v u m v m for some u i C. Now, 0 = <u, v i> = u i for each i = 1,..., r and also 0 = <u, u> = (u 1 ) (u m ) 2. So since r = m or r = m 1, u = 0, i.e. φ e j = 0 for each j = r + 1,..., n. Now, there is an isomorphism 6 α : C m C m which takes e 1,..., e r to v 1,..., v r (respectively). This is because the isomorphisms of C m act transitively on the set of ordered orthonormal bases of C m. (The m m matrix A = [v 1... v m] whose columns are the vectors v i acts on C m by taking e i to v i and A T A = I since the vectors v 1,..., v m are orthonormal). Hence φ = α φ and so φ = (ψ 1 h α ψ h) φ which completes the proof. Armed with Lemmas 1.6 and 1.8 we can now completely solve Problem 1.7. Theorem 1.9 For i, j = 1,..., n, let x ij C be given complex numbers so 4 We take C n and C m to be endowed with the bilinear forms which are represented by the diagonal matrices diag(1,..., 1, 0,..., 0) with rank(g) and rank(h) non-zero entries (respectively). 5 This follows from the fact that dim(vk+1 ) = dim(v k ) 1 (in the notation used at the start of this proof). 6 i.e. an isomorphism of C m endowed with the bilinear form represented by diag(1,..., 1) with respect to the standard basis e 1,..., e m. 25

27 that x ij = x ji, x ii = 1 and not all x ij = ±1. Then there exist l 1,..., l n SL for which <l i, l j > = x ij if and only if rank(x) 3, where X is the n n matrix whose (i, j) th entry is x ij. The arrangement of lines l 1,..., l n is unique up to the action of PSL 2 C {±1}, i.e. unique up to isometry and simultaneous reversal of orientations. Note that if all x ij = ±1 then the x ij may or may not be able to be realised by an arrangement of lines, and even if such an arrangement exists then it may not be unique. Proof of Theorem 1.9: Let x ij C be given complex numbers and suppose that there exist lines l 1,..., l n SL so that x ij = <l i, l j >. Then since L is 3-dimensional, any four of the lines must be linearly dependent. From this it follows that any four rows of X are linearly independent. For example, there exist constants α 1,..., α 4 C so that α 1 l α 4 l 4 = 0 and hence α 1 <l 1, l i > α 4 <l 4, l i > = 0 (i.e. α 1 x 1i α 4 x 4i = 0) for each i = 1,..., n. Hence rank(x) 3. So now suppose that for i, j = 1,..., n we have x ij C so that x ij = x ji and x ii = 1 and that rank(x) 3 where X = [x ij ] is the matrix defined in the statement. Define a symmetric, bilinear form g on C n by g(x, y) = x T Xy for any x, y C n. Then rank(g) 3 = rank(<, >) so setting (V, g) = (C n, g) and (W, h) = (L, <, >) in Lemma 1.8 gives us a linear map φ : C n L so that φ def <, > = g. For each i = 1,..., n let l i = φe i where e 1,..., e n is the standard basis for C n. Then x ij = e it Xe j = g(e i, e j ) = <φe i, φe j > = <l i, l j > for any i, j = 1,..., n. Note that <l i, l i > = x ii = 1 for each i = 1,... n so l i SL, i.e. each l i corresponds to an oriented line in H 3. Now, if two rows of X, say rows i and j, are linearly dependent then these two rows will be multiples of each other. But since x ii = x jj = 1 this means that x ij = x ji = ±1. So if we assume (as in the statement of the theorem) that there is some x ij ±1 then it follows that rank(g) = 2 or 3. Hence 26

28 by Lemma 1.8, φ (and hence the arrangement of lines l 1,..., l n ) is unique up to composing φ by an isomorphism of (L, <, >). But by Lemma 1.6 the isomorphisms of (L, <, >) are exactly given by the action of PSL 2 C {±1} on L. Hence the arrangement of lines l 1,..., l n is unique up to isometry and simultaneous reversal of orientations. The special cases of Theorem 1.9 when n = 3 and n = 4 are of particular interest to us. If three lines l 1, l 2, l 3 SL all have distinct end-points on S 2 then any two of these lines has a common-perpendicular. Adding these three perpendiculars and truncating appropriately gives an arrangement of six line segments in H 3 which meet at right-angles, i.e. a right-angled hexagon (see Figure 1.2(a)). Motivated by this generic case we will simply say that any three oriented lines define a right-angled hexagon, without making the assumption that the lines have distinct end-points. l 1 l 1 l2 l3 l2 l3 l 4 (a) A right-angled hexagon (b) A hextet Figure 1.2: Arrangements of (a) three lines and (b) four lines. Similarly, any four lines l 1,..., l 4 SL which all have distinct end-points on S 2 have six pair-wise common-perpendiculars. Adding these perpendiculars and truncating all lines appropriately gives an arrangement of lines in H 3 loosely resembling a tetrahedron (see Figure 1.2(b)). Since this arrangement is like a tetrahedron whose vertices have been stretched into the 27

29 lines l 1,..., l 4, turning its faces into right-angled hexagons, we call such an arrangement a hextet. As with right-angled hexagons, we will drop the requirement that the end-points of l 1,..., l 4 be distinct and simply say that any four oriented lines in H 3 define a hextet. Substituting n = 3 and n = 4 into Theorem 1.9 gives the following propositions. Proposition 1.10 (Existence and Rigidity of Right-Angled Hexagons) Let x ij C for i, j = 1, 2, 3 be given complex numbers so that x ij = x ji, x ii = 1 and not all x ij = ±1. Then there exist l 1, l 2, l 3 SL so that <l i, l j > = x ij and these lines are unique up to isometry and simultaneous reversal of the orientation of each line. Proposition 1.11 (Existence and Rigidity of Hextets) Let x ij C for i, j = 1,..., 4 be given complex numbers so that x ij = x ji, x ii = 1 and not all x ij = ±1. Then there exist l 1,..., l 4 SL so that <l i, l j > = x ij if and only if the x ij satisfy the hextet equation 1 x 12 x 13 x 14 x 21 1 x 23 x 24 0 = det x 31 x 32 1 x 34 x 41 x 42 x 43 1 = x 2 12x x 2 13x x 2 14x x 13 x 24 x 14 x 23 2x 12 x 34 x 14 x 23 2x 12 x 34 x 13 x 24. These lines l 1,..., l 4 are unique up to isometry and simultaneous reversal of each line s orientation. We finish this section with a brief discussion of degenerate and non-degenerate right-angled hexagons. 28

30 Definition 1.12 A right-angled hexagon defined by the lines l 1, l 2, l 3 SL is non-degenerate if l 1, l 2, l 3 SL form a basis for L and degenerate otherwise. If three lines l 1, l 2, l 3 all have a common perpendicular n then they all lie in the 2-dimensional subspace {L L <n, L> = 0} and so define a degenerate hexagon. It is not hard to prove that the converse is true if we also assume that <l i, l j > ±1 for some i, j = 1, 2, 3 (otherwise pair-wise common perpendiculars either will not exist or will not be unique). The following lemma gives two more characterisations of degenerate hexagons. Lemma 1.13 A right-angled hexagon defined by the lines l 1, l 2, l 3 SL is degenerate if and only if 1 x 12 x 13 0 = det x 21 1 x 23 x 31 x 32 1 where x ij = <l i, l j >. If some x ij ±1 then it is also true that the hexagon defined by lines l 1, l 2, l 3 hexagon defined by l 1, l 2, l 3. Proof: If α 1 l 1 + α 2 l 2 + α 3 l 3 = 0 then is degenerate if and only if it is isometric to the α 1 <l 1, l i > + α 2 <l 2, l i > + α 3 <l 3, l i > = 0 for each i = 1, 2, 3. Hence the rows of X def = 1 x 12 x 13 x 21 1 x 23 x 31 x 32 1 are linearly dependent and so det X = 0. Conversely, suppose that det X = 0 and assume (in order to derive a contradiction) that the l i are linearly independent. Then the rows of X are linearly dependent, so there are α 1, α 2, α 3 C so that <α 1 l 1 + α 2 l 2 + α 3 l 3, l i > = 0 29

31 for each i = 1, 2, 3. But since <, > is non-degenerate and (by assumption) l 1, l 2 and l 3 form a basis for L, this implies α 1 l 1 + α 2 l 2 + α 3 l 3 = 0 which is a contradiction. Now, suppose there is some g PSL 2 C so that g l i = l i for each i = 1, 2, 3. If l 1, l 2 and l 3 form a basis for L then g is an isometry which takes every oriented line to the same line but with the opposite orientation. This is absurd so the hexagon defined by l 1, l 2, l 3 must be degenerate. Conversely, suppose that x 12 ±1 and that there exist constants α 1, α 2, α 3 C so that α 1 l 1 + α 2 l 2 + α 3 l 3 = 0. Then since <l 1, l 2 > = x 12 ±1, l 1 ±l 2 and so (since l 1, l 2 SL) l 1 and l 2 are linearly independent. Hence α 3 0 and also l 1 and l 2 have a common perpendicular n SL (whose orientation is not unique). But then 0 = <α 1 l 1 + α 2 l 2 + α 3 l 3, n> = α 1 <l 1, n> + α 2 <l 2, n> + α 3 <l 3, n> = α 3 <l 3, n> where the last step follows by Corollary 1.5. Since α 3 0 this implies <l 3, n> = 0 which means that l 3 is also perpendicular to n (again see Corollary 1.5). So if g PSL 2 C acts as a half-turn about n then g l i = l i for each i = 1, 2, 3. 30

32 Chapter 2 Surfaces Equidistant from Two Lines In Section 3.1 we will study regions of hyperbolic 3-space H 3 which consist of all points closer to a central line Σ in H 3 than to any of the lines in a given collection. We will use a similar construction in Chapter 6 in an attempt to build incomplete hyperbolic structures on 3-manifolds. The boundary of such regions is contained in certain surfaces, each of which consists of points of H 3 which are equidistant from Σ and another line. We investigate such equidistant surfaces and their intersections in this chapter. We begin by briefly describing two models of hyperbolic 3-space, namely the hyperboloid model and the projective ball model (see Section 2.1). We then calculate explicit equations for each of these models which describe the equidistant surface between an arbitrary pair of lines (see Section 2.2). It follows from this calculation that the equidistant surface between a pair of disjoint lines is a quadratic, doubly-ruled surface homeomorphic to R 2. We then consider the intersection of two equidistant surfaces in the special case where the pairs of lines defining the surfaces have a line in common but are otherwise disjoint and share no end-points on the sphere at infinity (see Section 2.3). We show that such surfaces either coincide or meet in general position and further that their intersection contains no circles and has at most four components. 31

33 Contrary to the notation of Chapter 1, in this chapter <, > will denote the Minkowski metric and H 3 will usually denote the hyperboloid model or the projective ball model of hyperbolic 3-space. 2.1 Two Models of Hyperbolic 3-Space In this section we briefly recall some relevant facts about the hyperboloid and projective ball models of hyperbolic 3-space. For justifications and details, see Ratcliffe [40, ] or Thurston [46, 2.3]. Define Minkowski space E 1,3 to be R 4 endowed with the indefinite metric <, > given by <x, y> = x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 for any x, y E 1,3, where x = (x 0, x 1, x 2, x 3 ) and y = (y 0, y 1, y 2, y 3 ). Here we have used the vector space structure of R 4 to identify each each of the tangent spaces of E 1,3 with E 1,3 itself. The hyperboloid model of hyperbolic 3-space is the hyperboloid {x = (x 0, x 1, x 2, x 3 ) E 1,3 <x, x> = 1 and x 0 > 0} (2.1) equipped with the metric <, >. The restriction of <, > to (2.1) is positive definite and so defines a Riemannian metric on the hyperboloid. The Lorentz transformations are the linear transformations E 1,3 E 1,3 which preserve <, >. The Lorentz transformations in the connected component of the identity act on the hyperboloid (2.1) as the group of orientation-preserving hyperbolic isometries. Geodesics in the hyperboloid model are the (nonempty) intersections of the hyperboloid with the 2-planes of E 1,3 which pass through the origin. The projective ball model of hyperbolic 3-space is the 3-ball {(1, x 1, x 2, x 3 ) E 1,3 x x x 2 3 < 1} (2.2) equipped with the unique Riemannian metric for which the map from the 3-ball (2.2) to the hyperboloid model (2.1) given by projecting along lines 32

34 through the origin is an isometry. Therefore the geodesics of the projective ball model are the intersections of the 3-ball (2.2) with the 2-planes of E 1,3 which pass through the origin, and so they are straight lines with respect to the affine structure of E 1,3. There is an obvious embedding of the 3-ball (2.2) into R 3 given by (1, x 1, x 2, x 3 ) (x 1, x 2, x 3 ). We will usually use this embedding to identify the projective ball model with the 3-ball B = {(x 1, x 2, x 3 ) R 3 x x x 2 3 < 1}. Under this identification, the group of hyperbolic isometries corresponds to the group of projective transformations of R 3 which preserve B. In either model, the distance d H 3(x, y) between two points x, y H 3 is defined in the usual way for a Riemannian manifold as the length of the shortest path between x and y. 2.2 Explicit Equations for Equidistant Surfaces In this section we will find algebraic equations which (in the hyperboloid and projective ball models of H 3 ) describe the locus of points which are equidistant from a given pair of lines. We will then use these explicit descriptions to prove that the equidistant surface between two disjoint lines is diffeomorphic to R 2. As remarked in Section 2.1, a line l in the hyperboloid model is the intersection of the hyperboloid (2.1) with some 2-dimensional subspace V l of E 1,3. Let a, b E 1,3 span V l and suppose that a and b are normalised so that <a, a> = 1, <b, b> = 1, <a, b> = 0. (2.3) For instance, we could take b to be a point of l and a to be a unit tangent vector to l at b. Now, let x be any point of the hyperboloid (2.1). The distance d H 3(x, l) from x to l is d H 3(x, l) def = inf{d H 3(x, y) y l} 33

35 and it is a standard fact that d H 3(x, l) = d H 3(x, p) for some unique point p of l. Hence p is a scalar multiple of the orthogonal projection 1 π(x) of x onto V l. To see this, note that p is the unique point of l which is fixed by the reflection r of H 3 in the plane which passes through x and is perpendicular to l. In the hyperboloid model, r is achieved by a Lorentz transformation r : E 1,3 E 1,3 which fixes x and takes V l to itself. Hence r also fixes π(x), so the scalar multiple of π(x) which lies in the hyperboloid (2.1) is therefore p. Now, π(x) = sa + tb for some s, t R and π(x) x is perpendicular to V l, i.e. <sa + tb x, a> = 0 and <sa + tb x, b> = 0. Using the normalisations (2.3) these equations become Hence π(x) = <x, a>a <x, b>b. s <x, a> = 0 and t <x, b> = 0. Now, <π(x), π(x)> = <x, a> 2 <x, b> 2 and it is not hard to see that this is always strictly less than zero 2. Hence up to sign p is <x, a>a <x, b>b <x, b>2 <x, a> 2. (2.4) It is not hard to see that p is actually equal to (2.4), but since we don t need this result we will not prove it here. Now, the distance d H 3(u, v) between two points u and v of the hyperboloid model is given by cosh(d H 3(u, v)) = <u, v> (see [40, 3.2]). Hence cosh 2 (d H 3(l, x)) = cosh 2 (d H 3(p, x)) = <p, x> 2 (2.5) = (<x, a>2 <x, b> 2 ) 2 <x, b> 2 <x, a> 2 = <x, b> 2 <x, a> 2. (2.6) 1 The orthogonal projection of x E 1,3 onto V l is the unique point π(x) V l so that π(x) x is perpendicular to every point of V l. 2 There is a Lorentz transformation taking taking a to (0, 1, 0, 0), b to (1, 0, 0, 0) and x to some point y = (y 0, y 1, y 2, y 3 ) of the hyperboloid (2.1) (since the Lorentz transformations act transitively on the orthonormal frames in E 1,3 ). Hence <x, a> 2 <x, b> 2 = y 2 1 y 2 0 = 1 y 2 2 y 2 3 < 0. 34

36 Since cosh 2 ( ) is strictly monotonic on the non-negative real numbers, this proves the following proposition. Proposition 2.1 Let l 1 and l 2 be two lines in the hyperboloid model of H 3. Suppose that l 1 and l 2 lie in the subspaces of E 1,3 spanned by a 1, b 1 E 1,3 and a 2, b 2 E 1,3 (respectively), and that these are normalised so that <a 1, a 1 > = 1, <b 1, b 1 > = 1, <a 1, b 1 > = 0 and <a 2, a 2 > = 1, <b 2, b 2 > = 1, <a 2, b 2 > = 0. Then the equidistant surface between l 1 and l 2 is the intersection of the hyperboloid (2.1) with the hypersurface of points x E 1,3 which satisfy <x, a 1 > 2 <x, b 1 > 2 = <x, a 2 > 2 <x, b 2 > 2. (2.7) Note that a 1, b 1 and a 2, b 2 also define two lines in the the projective ball model (see Section 2.1). Then by Proposition 2.1, the equidistant surface between these two lines is the intersection of the hypersurface (2.7) with the 3-ball (2.2). This is because equation (2.7) is homogeneous and because the isometry from the 3-ball (2.2) to the hyperboloid (2.1) is the projection from the origin. The following proposition gives the distance from a point to a line in terms of the end-points of the line. Proposition 2.2 For a point x and a line l in the hyperboloid model (2.1) of H 3, 1 2 cosh2 d H 3(x, l) = <x, u><x, v> <u, v> where l lies in the span of u, v E 1,3 and <u, u> = <v, v> = 0. Proof: The proof is almost identical to the proof of Proposition 2.1 given above. Given two lines l 1 and l 2 in H 3 we define the unoriented complex distance between l 1 and l 2 to be the complex distance d C (ˆl 1, ˆl 2 ) (see Definition 1.2) 35

37 between ˆl 1 and ˆl 2, where ˆl 1 and ˆl 2 are l 1 and l 2 (respectively) endowed with arbitrary orientations. Changing the orientation on one of ˆl 1 or ˆl 2 has the effect of adding iπ to d C (ˆl 1, ˆl 2 ), so the unoriented complex distance is only well-defined up to sign and an additive iπz indeterminacy (c.f. Definition 1.2). Consider the two lines which lie in the span of a 1, b 1 E 1,3 and a 2, b 2 E 1,3 where and a 1 = (0, cos θ 2, sin θ 2, 0), and b 1 = (cosh δ 2, 0, 0, sinh δ 2 ) (2.8) a 2 = (0, cos θ 2, sin θ 2, 0), and b 2 = (cosh δ 2, 0, 0, sinh δ 2 ) (2.9) for some δ, θ R. Note that a 1, b 2 and a 2, b 2 are normalised as in Proposition 2.1 and that the lines given by (2.8) and (2.9) are an unoriented complex distance δ + iθ apart. Now, <x, a 1 > 2 <x, a 2 > 2 = ( x 1 cos θ 2 x 2 sin 2) θ 2 ( x 1 cos θ 2 + x 2 sin θ ) 2 2 = 4x 1 x 2 cos θ 2 sin θ 2 = 2x 1 x 2 sin θ and <x, b 2 > 2 <x, b 2 > 2 = ( x 0 cosh δ 2 + x 3 sinh 2) δ 2 ( x 0 cosh δ 2 x 3 sinh δ ) 2 2 = 4x 0 x 3 sinh δ 2 cosh δ 2 = 2x 0 x 3 sinh δ. So in the hyperboloid model of H 3, the equidistant surface between the lines given by (2.8) and (2.9) is the intersection of the hyperboloid (2.1) with the hypersurface given by x 1 x 2 sin θ = x 0 x 3 sinh δ (2.10) 36

38 for any x = (x 0, x 1, x 2, x 3 ) E 1,3 (by Proposition 2.1). Hence in the projective ball model B, the equidistant surface between the lines given by (2.8) and (2.9) is the intersection of B with the hypersurface x 1 x 2 sin θ = x 3 sinh δ (2.11) for any x = (x 1, x 2, x 3 ) B, by the comment following Proposition 2.1. Now, let l 1 and l 2 be two lines in the projective ball model B of H 3 which share no end-points on the sphere at infinity. Then if l 1 and l 2 are an unoriented distance δ + iθ apart then there is an (orientation-preserving) isometry of H 3 taking l 1 and l 2 to the lines defined by (2.8) and (2.9). (This follows from Theorem 1.9 and the fact that cosh(δ + iθ) ±1 since l 1 and l 2 share no end-points on the sphere at infinity.) Hence the equidistant surface between l 1 and l 2 is isometric to the intersection of B with the surface given by (2.11). Note that if we replace δ + iθ by δ + iθ + iπ then the right-hand side of (2.11) changes sign while the left-hand side does not. This causes no problems, however, since the equations x 1 x 2 sin θ = x 3 sinh δ and x 1 x 2 sin θ = x 3 sinh δ represent the same surface in H 3 up to orientation-preserving isometry. Now, if the lines l 1 and l 2 are disjoint then the equidistant surface between them is diffeomorphic to R 2. This follows from the fact that if l 1 and l 2 are disjoint 3 then δ 0 so (2.11) becomes x 3 = (sin θ/ sinh δ)x 1 x 2. A simple calculation shows that any surface given by an equation of the form x 3 = kx 1 x 2 for some k R is transverse to the (ideal) boundary S 2 = {(x 1, x 2, x 3 ) R 3 x x x 2 3 = 1} of B. Since the intersection of S 2 and x 3 = kx 1 x 2 is clearly a circle when k = 0, this implies that the intersection of S 2 and x 3 = kx 1 x 2 is a (topological) circle for all k R. Since the projection of the surface x 3 = kx 1 x 2 onto the x 1 -x 2 plane is a diffeomorphism, this implies that the part of the surface x 3 = kx 1 x 2 contained in B is diffeomorphic to R 2. Hence the equidistant surface between l 1 and l 2 is diffeomorphic to R 2 as claimed. The surfaces in R 3 given by the equation x 3 = kx 1 x 2 for some k R are called hyperbolic paraboloids. These quadratic surfaces have many nice properties, 3 Recall that we are already assuming l 1 and l 2 do not share an end-point on the sphere at infinity. 37

39 for instance they are doubly-ruled. Since straight lines in R 3 are hyperbolic lines in the projective ball model, this ruling is by hyperbolic lines. This proves the following proposition. Proposition 2.3 Let l 1 and l 2 be two lines in the projective ball model B of H 3 which do not share an end-point on the sphere at infinity and let δ + iθ be the unoriented complex distance between them. Then the equidistant surface between l 1 and l 2 is isometric to the set of all points x = (x 1, x 2, x 3 ) B which satisfy x 1 x 2 sin θ = x 3 sinh δ. If l 1 and l 2 do not intersect (i.e. if δ 0) then the equidistant surface between l 1 and l 2 is diffeomorphic to R 2 and has a double ruling by hyperbolic lines. Note that since the isometries of the projective ball model B are a subgroup of the projective isometries of R 3, if δ 0 then the equidistant surface of Proposition 2.3 is projectively equivalent to the part of some surface x 3 = kx 1 x 2 contained inside B. This implies that the equidistant surface between l 1 and l 2 is either a hyperbolic paraboloid or a hyperboloid of one sheet. 2.3 Intersections of Equidistant Surfaces In Section 3.1 we will consider a region of H 3 whose boundary consists of equidistant surfaces, each of which is equidistant from a central line Σ and one other line which is disjoint from Σ and shares no end-points with it. In the present section we will show that any two of these equidistant surfaces are in general position and that their intersections can contain at most four topological lines and no circles. It is worth noting that the equidistant surfaces between two arbitrary pairs l 1, l 2 and l 3, l 4 of lines in H 3 can intersect in a singular variety or with circular components. It is only under the additional assumption l 1 = l 4 that the equidistant surfaces between the pairs l 1, l 2 and l 3, l 4 have the intersection properties quoted above. 38

40 We first establish some notation for this section. Let l 1, l 2 and l 3 be three disjoint (unoriented) lines in H 3 which have no end-points in common on S 2. Then for any distinct i, j = 1, 2, 3 there exists a unique (unoriented) common perpendicular between l i and l j (see Section 1.1) which we denote l ij (hence l ij = l ji ). Define functions d i : H 3 R by d i = d H 3(l i, ) and let d ij = d i d j. Let F ij denote the equidistant surface (as described in Section 2.2) between l i and l j. Then F ij = F ji is the set of points x H 3 which satisfy d ij (x) = 0. Note that the intersection of any two of the equidistant surfaces F ij is actually equal to the intersection F 12 F 23 F 31 of all of them. (For instance, if x F 12 F 23 then d 1 (x) = d 2 (x) and d 2 (x) = d 3 (x) so d 1 (x) = d 3 (x) and hence x F 31.) The hyperbolic metric g(, ) on H 3 defines an isomorphism between vector fields ξ on H 3 and 1-forms ω on H 3 given by the formula ω( ) = g(ξ, ). (2.12) Given any smooth real-valued function φ defined on an open subset of H 3 we define the gradient φ of φ by taking the exterior derivative dφ of φ and then applying the isomorphism (2.12) from 1-forms to vector fields. For each i = 1, 2, 3, d i is a smooth function on H 3 \ l i, so d i is a smooth vector field defined on H 3 \ l i. The following elementary observations are the key ingredients in the proofs (below) that any two of the equidistant surfaces F ij meet in general position and do not contain any circles of intersection. Lemma 2.4 For any distinct i, j = 1, 2, 3, the following hold: D1 d i is a unit vector field. D2 d i always points directly away from l i, i.e. at each point x H 3 \ l i, d i is tangential to the line which passes through x and is perpendicular to l i. D3 d ij is zero only on the extremities of l ij, i.e. only on the parts of l ij not between l i and l j, as illustrated in bold in Figure

41 D4 The restriction of d ij to F ij is a nowhere-zero normal vector field to the surface F ij. D5 If d 12 and d 13 are proportional at some point then d ij = 0 at that point for some distinct i and j. l i l j li j Figure 2.1: The extremities of l ij Proof: While (D1) and (D2) are true from quite general principles, we prove them here with a simple calculation. In the upper half-space model {(z, t) C R t > 0} (see Section 1.1) take l i to be the line with end-points 1 and 1 on S 2 and consider a point x H 3 of the form (0, t) C R with t > 1. Clearly d i is invariant under any isometry of H 3 which fixes l i. There are two reflections which preserve l i and the point x and these only fix vectors at x which are tangential to the t-axis of the upper-half space. Hence d i = λ t at x for def some λ R, where t =. t Now, from the definition of the metric on the upper half-space model (see [46, 2.2]) t t is a unit vector at x and a simple calculation shows that d i (x) = log t. Hence g( d i, t t ) = dd i (t t ) by (2.12) = t d i by the definition of d t = 1 since d i (x) = log t. 40

42 But also g( d i, t t ) = g(λ t, t t ) = λ/t so λ = t and hence d i = t t. These computations prove (D1) and (D2). Now, d ij = d i d j is zero when d i = d j which by (D2) only occurs on the common perpendicular l ij to l i and l j. However, on l ij between l i and l j, d i = d j 0 so so (D3) is proved. Since F ij is a level set of d ij, d ij is perpendicular to F ij at all points of F ij. But d ij is non-zero at every point of F ij by (D3), since the only point of l ij which lies on F ij is the point half-way between l i and l j. To prove (D5), note that at any point x H 3, d 1 x, d 2 x and d 3 x are points on the unit sphere of the tangent space to H 3 at x. So in this tangent space, d 12 x and d 13 x point in the same direction as the lines which pass through d 1 x, d 2 x and d 1 x, d 3 x (respectively). Hence if d 12 x and d 13 x are proportional then these lines are parallel. Since both lines pass through the point d 1 x, they must coincide. But a line can meet a sphere in at most two points, so two of the d i x must be equal. Hence d ij x = 0 for some i j. It is now easy to prove the following lemma. Lemma 2.5 Let l 1, l 2, l 3 be three disjoint lines in H 3 which don t share any end-points on S 2, and for each i j let F ij be the equidistant surface between l i and l j. Then F 12 and F 13 are in general position, so F 12 F 13 is either empty or a smooth 1-manifold. Further, at most four connected components of F 12 F 13 are diffeomorphic to lines. Proof: F 12 and F 13 are transverse at all points of F 12 F 13 because by (D4) and (D5) of Lemma 2.4, if the normals to F 12 and F 13 are proportional at any point then d ij = 0 at that point (for some i, j). But (D4) implies that each d ij is non-zero at the points of the intersection F 12 F 13 = F 12 F 13 F 23, so F 12 and F 13 must be in general position. Hence F 12 F 13 is a smooth 1-manifold (without boundary) whose connected components are therefore (topological) circles and lines. The lines are closed subsets of H 3 but topologically they are open intervals, so they must extend 41

43 out to the boundary S 2 of H 3. Hence any sufficiently large sphere 4 which is contained in the projective ball model will intersect F 12 F 13 in twice the number of points as there are components of F 12 F 13. But in the projective ball model, the sphere and each equidistant surface is a quadratic surface (by Proposition 2.3) so by Bezout s theorem (see [19] or [42]) there are at most eight points in their intersection. Hence there are at most four topological lines in F 12 F 13. The rest of this section is devoted to proving Proposition 2.6 (below). The proof of this proposition will involve counting the zeroes of a vector field, so we pause briefly to recall some definitions (e.g. see Milnor [33, 6]). Let ξ be a vector field on R n which has an isolated zero at x R n. Since R n is a vector space, the tangent spaces to R n can all be canonically identified with R n itself. Under this identification, ξ becomes a smooth map ξ : R n R n. We say that the zero of ξ at x is non-degenerate if ξ is a local diffeomorphism at x. A non-degenerate zero of ξ at x has index +1 if ξ : R n R n is locally orientation-preserving at x and index 1 if ξ is locally orientation-reversing. It is also possible to define the index of a degenerate, isolated zero of ξ, though we have no use of this definition at this stage. The notions of non-degeneracy and index extend to the zeroes of a vector field on a smooth n-manifold M by using co-ordinate charts about the isolated zeroes of the vector field to obtain a corresponding vector field on R n. Since the non-degeneracy and the index of each zero are independent of the choice of co-ordinate chart (see [33, 6]), these notions are well defined on M. Proposition 2.6 Let l 1, l 2, l 3 be three disjoint lines in H 3 which don t share any end-points on S 2, and for i j let F ij be the equidistant surface between l i and l j. Then F 12 F 13 is a smooth 1-manifold with at most four connected components, each of which is diffeomorphic to a line. By Lemma 2.5, the proposition will be proved if we can show that there are no smooth circles in the intersection of F 12 and F 13. So assume (in order to 4 The point of taking a sphere other than than S 2 is to ensure that the end-points on S 2 of each line of intersection are distinct, though this probably is true if we just take S 2 itself. 42

44 prove a contradiction) that there is a circular component of F 12 F 13. By Proposition 2.3, F 23 is diffeomorphic to R 2 so this circle of intersection is the boundary D of a disc D contained in the plane F 23. Define ρ : F 23 R to be the restriction of d 12 or d 13 to F 23 (note that these restrictions are equal). Let ρ be the gradient of ρ with respect to the restriction of the hyperbolic metric to F 23. Then ρ is a vector field whose vectors are tangential to F 23 and which is defined at every point of F 23 where d 12 or d 13 are smooth. Then by Lemma 2.8 (below), ρ is defined and continuous in a neighbourhood of D, and by Lemma 2.7 (also below) it points outwards 5 at each point of D. So by applying the Poincaré-Hopf Theorem to ρ on D (see [33, 6] or [5]), the sum of the indices of the zeroes of ρ in D should equal 1. However, by Lemma 2.9 (below) all zeroes of ρ are isolated and have index 1, so we have a contradiction. Hence Proposition 2.6 is proved once we have established the following three lemmas. Lemma 2.7 The vector field ρ is non-zero on D and perpendicular to D everywhere. Proof: First note that ρ = (d 12 F 23 ) is the orthogonal projection of d 12 onto F 23. Hence it is zero only where d 12 is normal to F 23. But by (D4) and (D5) of Lemma 2.4 this can only occur when d ij = 0 for some i, j. But by (D4) this cannot occur at any point of D F ij, so ρ is non-zero at each point of D. Now, note that F 12 F 13 F 23 is exactly the subset of F 23 where ρ : F 23 R is zero so D is contained in a level set of ρ. Hence ρ is everywhere perpendicular to D. Lemma 2.8 The vector field ρ is defined and smooth on a neighbourhood of D in F Actually, Lemma 2.7 shows that either ρ or ρ points outwards at D. By multiplying ρ by 1 if need be we can ensure it points outwards. Note that since D is 2-dimensional, replacing ρ by ρ doesn t affect the indices of the zeroes of ρ. 43

45 Proof: We will prove that d 1, d 2 and d 3 are all smoothly defined in a neighbourhood (in H 3 ) of D. Since ρ = (d 12 F 23 ) is the orthogonal projection of d 12 = d 1 d 2 onto F 23, this will prove the lemma. But since d i is smoothly defined at all points of H 3 \ l i, this is equivalent to showing that l 1, l 2 and l 3 do not intersect D. Now, a line l of one of the rulings of F 23 (see Proposition 2.3) which passes through an interior point p of D will meet D in at least two points. Hence l also meets F 12 in at least two points (since D F 12 ) but by degree considerations (i.e. by Bezout s theorem (see [19] or [42]) and Proposition 2.3) this is the maximum. Hence there can be no points of F 12 contained in the interior of D (since taking p to be such a point would yield a contradiction). By the same reasoning there can be no points of F 13 in the interior of D, either. This shows that there are no nested circled of intersection on F 23 (or on any of the other surfaces). Now, by Proposition 2.3, D is the boundary of the disc D F 12 and it is also the boundary of a disc D F 13. We have just shown that D only meets F 13 and F 23 on its boundary (which is equal to D) and that D only meets F 12 and F 23 on its boundary (which is also equal to D). Hence the union of any two of D, D and D form the boundary of an embedded piecewise smooth sphere in H 3, so by the Schönflies theorem (see Alexander [1] or Brown [7]) these spheres each bound a 3-ball. One of these 3-balls B is the union of the other two, and after possibly permuting the indices of l 1, l 2, l 3 we can assume that B = D D, i.e. that the interior of D is contained in the interior of B (see Figure 2.2). As noted above, to prove the lemma we just have to show that l 1, l 2 and l 3 don t meet D at any point. But it is clear that l 2 and l 3 cannot meet D because l 2 and l 3 are disjoint and D lies in the equidistant surface between them. Similarly, l 1 is disjoint from both F 12 and F 13. So l 1 cannot meet B (because B = D D is contained in the union of F 12 and F 13, see Figure 2.2). But l 1 is unbounded so it cannot meet an interior point of B without passing through B. Hence l 1 cannot meet B at all, so l 1 cannot meet D B. Lemma 2.9 The zeroes of ρ are all isolated and have index 1. Proof: As noted above, ρ = (d 12 F 23 ) is the orthogonal projection of d 12 44

46 D F 13 F 23 D F 12 D Figure 2.2: A neighbourhood of the ball B in H 3. onto F 23. Hence ρ = 0 only where d 12 is perpendicular to F 23. But by (D4) and (D5) of Lemma 2.4, this only occurs when d ij = 0 for some i, j. But by (D4) again, d 23 0 at any point of F 23. Hence by (D3), the zeroes of ρ can only be at points of intersection between F 23 and the extremities of l 12 or l 13. We will calculate an expression for d 1 and d 2 at an arbitrary point p of l 12 up to second order in our local coordinates. From this we will be able to calculate ρ to first order. From this we will be able to show that any zero of ρ is non-degenerate and of index 1. Consider the projective ball model B R 3 of H 3 (see Section 2.1). We can assume without loss of generality that: p is the origin; l 12 is the x 3 -axis; l 1 lies in the intersection of the planes (x 3 = tanh δ 1 ) and (x 2 = 0) for some δ 1 > 0; and l 2 lies in the plane (x 3 = tanh δ 2 ) for some δ 2 > 0 and makes an angle θ to the plane (x 2 = 0) (see Figure 2.3). In order to calculate d 1 about p, briefly consider the ball B to lie in the (x 0 = 1) hyperplane of E 1,3 (see Section 2.1). Then l 1 lies in the span of a = (0, 1, 0, 0) and b = (cosh δ 1, 0, 0, sinh δ 1 ). Note that these vectors are normalised so that <a, a> = 1, <b, b> = 1 and <a, b> = 0. Now, under the isometry between the hyperboloid model and the projective ball model (i.e. projection from the origin), l 1 corresponds to the line (also denoted l 1 ) in the hyperboloid (2.1) defined by a and b. For any x in the hyperboloid 45

47 x 3 x 2 p x 1 δ 1 δ 2 l 1 θ l 2 l 12 Figure 2.3: The set-up for Lemma 2.9 (2.1), <x, x> = 1 so by (2.6), cosh 2 d 1 (x) = <x, a>2 <x, b> 2. (2.13) <x, x> The right-hand side of (2.13) is clearly invariant under projection from the origin. But since this projection is the isometry between B and the hyperboloid (2.1), the left-hand side of (2.13) is also invariant. Hence the formula (2.13) holds for any point x = (1, x 1, x 2, x 3 ) of the projective ball model. Upon substituting the a and b given above and carrying out Taylor expansions we see that up to second order in the x i, d 1 (x) = δ 1 + x 3 + tanh δ 1 x 2 1/2 + coth δ 1 x 2 2/ (2.14) Note that the first two terms come from the lowest order term in the expansion of tanh(δ 1 + x 3 ), and the quadratic terms reflect the fact that the principal extrinsic curvatures of the boundary of a tube of constant radius δ 1 about l 1 are tanh δ 1 and coth δ 1. From (2.14) we immediately have that up to second order, d 2 (x) = δ 2 + x 3 + tanh δ 2 (x 1 cos θ + x 2 sin θ) 2 /2 + coth δ 2 ( x 1 sin θ + x 2 cos θ) 2 / (2.15) 46

48 Now, when l 12 intersects F 23 it does so transversely. This is because otherwise, by (D2) and (D4) of Lemma 2.4, d 2 is perpendicular at that point to the normal vector to the surface, d 23 = d 2 d 3. But by (D1), d 2 and d 3 are unit vectors, so d 2 = d 3. Hence d 23 = 0, contradicting (D4), so l 12 and F 23 must be in general position. Hence the zeroes of ρ are isolated. We now consider the projective ball model to be the unit ball B in R 3. Since F 23 is transverse to the x 3 -axis at 0, the projection of F 23 onto the x 1 -x 2 plane R 2 is a local diffeomorphism near the origin. The inverse of this projection is an embedding i : R 2 R 3 given by i : (x 1, x 2 ) (x 1, x 2, f(x 1, x 2 )) for some smooth f defined in a neighbourhood of 0 R 2. Now, ρ is defined to be the vector field on F 23 corresponding to dρ under the isomorphism Φ g between 1-forms and vector fields coming from the metric g on F 23 (which is the restriction of the hyperbolic metric to F 23 ). Via the diffeomorphism i we can pull ρ back to a vector field (denoted i ( ρ)) on R 2 which (by definition) has the same index as ρ. Then i ( ρ) is the vector field on R 2 corresponding to d(ρ i) under the isomorphism between 1-forms and vector fields on R 2 coming from the metric i g (the pull-back of g with respect to i), i.e. i ( ρ) = Φ i g(d(ρ i)). It is easy to calculate ρ i to second order in x 1, x 2 since by (2.14) and (2.15) d 1 d 2 has no x 3 dependence (to second order), and hence (x 1, x 2 ) i (x 1, x 2, f(x 1, x 2 )) ρ δ 1 δ 2 + tanh δ 1 x 2 1/2 tanh δ 2 (x 1 cos θ + x 2 sin θ) 2 /2 + coth δ 1 x 2 2/2 coth δ 2 ( x 1 sin θ + x 2 cos θ) 2 /2 +...(2.16) Now, for any 1-form ω and any two Riemannian metrics g 0 and g 1 on a smooth manifold, the vector fields Φ g0 (ω) and Φ g1 (ω) have the same zeroes and the same indices at each isolated zero. (This is because the Riemannian metrics on a given manifold form a path connected space, and because the index of a zero is invariant under homotopies which leave the zero isolated see [33, 6]). So if e is the usual Euclidean metric on R 2 then Φ e (d(ρ i)) has a zero at the origin with the same index as the zero of i ( ρ). As remarked in the definition of index given above, we can view the vector field Φ e (d(ρ i)) as a map V : R 2 R 2. Then V is a local diffeomorphism 47

49 at the origin if and only if the Jacobian determinant JV 0 of V evaluated at the origin is non-zero, and V is locally orientation-reversing at the origin if and only if JV 0 < 0. From (2.16) we know that up to first order: V (x 1, x 2 ) = ( cos θ tanh δ 2 (x 1 cos θ + x 2 sin θ) so we can calculate JV 0 exactly as + sin θ coth δ 2 ( x 1 sin θ + x 2 cos θ) + tanh δ 1 x , sin θ tanh δ 2 (x 1 cos θ + x 2 sin θ) cos θ coth δ 2 ( x 1 sin θ + x 2 cos θ) + coth δ 1 x ) JV 0 = 2 cos 2 θ(tanh δ 1 coth δ 2 + tanh δ 2 coth δ 1 ) sin 2 θ(tanh δ 1 tanh δ 2 + coth δ 1 coth δ 2 ). Now, an elementary argument shows that if t is a positive real number then t + 1/t 2 with equality exactly when t = 1. So as long as δ 1 δ 2 then tanh δ 1 coth δ 2 + tanh δ 2 coth δ 1 > 2 and tanh δ 1 tanh δ 2 + coth δ 1 coth δ 2 > 2 so JV 0 < 2 2 cos 2 θ 2 sin 2 θ = 0 which completes the proof of the lemma. 48

50 Chapter 3 Tube Domains Given a compact 3-manifold X whose boundary X consists of one or more 2-tori, a (topological) Dehn filling of X is the closed 3-manifold which results from gluing solid tori to the boundary of X. The importance of this construction comes from the fact that any closed 3-manifold can be expressed as the Dehn filling of the complement of a link in S 3 (see Lickorish [29] and Wallace [47]). If the interior of X admits a complete, finite-volume hyperbolic structure then there is a geometric version of this notion of Dehn filling due to Thurston which roughly consists of deforming the hyperbolic structure on the interior of X and then taking the metric completion (see [44]). The resulting space is said to be a (generalised) hyperbolic Dehn filling of X. In this chapter we consider a space M which is a hyperbolic Dehn filling of an orientable 3-manifold whose boundary is a single 2-torus. There is a distinguished compact set Σ in M coming from the filling 1 and M has a hyperbolic structure which is smooth everywhere except possibly at Σ, where certain types of singularities may occur. Corresponding to M and Σ we will define a canonical tube domain D Σ for M. If M is a smooth hyperbolic 3-manifold or if M is a cone-manifold 2 (with 1 If M is a Dehn filling of X then Σ is the set of points which are needed to complete the interior of X. 2 The results of this chapter are not limited to these two cases, though this restriction makes the results slightly easier to state. 49

51 arbitrary cone-angle) then D Σ is homeomorphic to a solid torus and it has a natural hyperbolic structure which may be singular along an unknotted core circle. Then M can be recovered from D Σ by isometrically identifying the faces 3 of D Σ in pairs. Some applications of tube domains include: (1) they (generically) give an algorithm for finding an ideal triangulation of the complement of a simple closed geodesic in a hyperbolic 3-manifold M (see Section 3.5) and (2) that they provide the theoretical underpinnings which allow us to calculate the tube radii of all but a finite number of (topological) Dehn fillings of a 3- manifold which admits a complete hyperbolic structure (see Section 4.6). We begin this chapter by considering the illustrative case when the hyperbolic structure of M is non-singular along Σ (see Section 3.1). We then consider the general case of Dehn surgery-type singularities along Σ, proving some basic facts about the geodesics of M (in Section 3.2) before defining the tube domain in general (see Section 3.3) and eliciting some of its properties (see Section 3.4). We finish by showing that the pair (M, Σ) determines a canonical spine for M \ Σ which, under certain genericity conditions, is dual to a canonical ideal triangulation for M \ Σ (see Section 3.5). 3.1 Simple Closed Geodesics in Closed Hyperbolic 3-Manifolds In this section we describe the tube domain of a (smooth) hyperbolic 3- manifold M about one of its simple closed geodesics Σ. The rationale behind devoting a section to this special case is that this case is interesting in its own right and it also illustrates the essential geometrical ideas behind the definition of the tube domain without being cluttered by technicalities. Let M be a closed, orientable hyperbolic 3-manifold and let Σ M be a simple closed geodesic 4. Then the universal cover of M is H 3 so we can 3 The faces of D Σ are certain subsets of the boundary of D Σ. 4 The results of this section are also valid if we take M to be a closed, orientable hyperbolic 3-orbifold whose singular set is a simple closed geodesic Σ, but for simplicity 50

52 identify M with the quotient of H 3 by a discrete group Γ of orientationpreserving hyperbolic isometries. We denote the corresponding covering projection π : H 3 M. Covering Σ is a collection lift(σ) of disjoint lines in H 3, no two of which share an end-point on the sphere at infinity. Choose a line Σ lift(σ) and define D Σ def = {x H 3 d H 3(x, Σ) d H 3(x, Σ ) Σ lift(σ)} (3.1) (where the distance d H 3(x, l) between a line l and a point x is the minimum of the distance between x and any point on l). Clearly the stabiliser Γ Σ of Σ Γ Σ = {g Γ g Σ = Σ} acts on D Σ, and we define the tube domain D Σ of M based at Σ to be def D Σ = D Σ/Γ Σ. Note that (up to isometry) D Σ is independent of our choice of Σ lift(σ) because Γ acts transitively on lift(σ). Lemma 3.1 There exist Σ 1,..., Σ m lift(σ) and a ball B H 3 so that D Σ B = {x B d H 3(x, Σ) d H 3(x, Σ i ) for every i = 1,..., m} (3.2) and D Σ Γ Σ B. Note that an immediate consequence of this lemma is that D Σ = {x H 3 d H 3(x, Σ) d H 3(x, g Σ i ) g Γ Σ i = 1,..., m}. Proof of Lemma 3.1: We first note that any ball B in H 3 only meets a finite number of the lines of lift(σ). Since Σ is a simple closed curve in M, there we restrict our attention to smooth hyperbolic 3-manifolds. If we took M to be a 1- cusped, orientable, finite-volume hyperbolic 3-manifold and Σ to be an embedded horoball neighbourhood of the cusp then D Σ (as defined later in this section) is the Ford domain of M (see [11, 4]) and many of the arguments given in this section are valid in this case, too. 51

53 is some ɛ > 0 so that an ɛ-neighbourhood of Σ is an embedded solid torus and so the (real) distance 5 between any two lines of lift(σ) is greater than 2ɛ. Now, for each line Σ lift(σ) which meets B, choose some point of Σ B (e.g. the point of Σ which is closest to the centre of B ) and consider the collection of balls of radius ɛ about these points. All of these balls are disjoint and they are all contained in an ɛ-neighbourhood of B. Hence by volume considerations there can be at most a finite number of such balls and so there can only be a finite number of lines in lift(σ) which meet B. Now choose any point p of Σ and let p be a point of Σ covering 6 p. Let B and B be the closed balls of radius D + l/2 and 2D + l (respectively) about p, where D is the diameter 7 of M and l is the (real) length of Σ. As argued above, B only meets a finite number of lines Σ 1,..., Σ m of lift(σ). Now, suppose that Σ lift(σ) is a line for which there exists some x B so that d H 3(x, Σ ) d H 3(x, Σ). Then by the triangle inequality, Σ must meet B and so Σ = Σ i for some i = 1,..., m. This proves (3.2). But now, for any x D Σ, d H 3(x, Σ) < D, and for any point x of Σ, d H 3(x, Γ Σ p) < l/2. Hence d H 3(x, Γ Σ p) < D + l/2 and so D Σ Γ Σ B. It follows from Lemma 3.1 that D Σ is homeomorphic to a solid torus. However, since we don t need this fact in this section, and since we prove it more generally in Lemma 3.19, we will skip the proof for now. Now, for any Σ lift(σ) so that Σ Σ, let E Σ H 3 be the equidistant surface between the two lines Σ and Σ, as discussed in Chapter 2 (see Propositions 2.3 and 2.6). If the interior of E Σ D Σ is non-empty then it consists of one or more connected components and we define the closure of each of these components to be a face of D Σ. We also define the images of the faces of D Σ under the covering projection D Σ D Σ to be the faces of D Σ. We will prove in Lemma 3.18 that D Σ has only a finite number of faces and that these faces cover D Σ and have disjoint interiors. Given Lemma 3.1 this should seem fairly plausible to the reader, so we again skip the proof here. 5 The real distance between two lines l and l is the minimum of d H 3(x, x ) over all points x l and x l. 6 i.e. π( p) = p where π : H 3 M is the covering projection. 7 The diameter of M is the maximum of d M (x, x ) for any x, x M. 52

54 Now, there is a unique map exp Σ : D Σ M which makes the diagram D Σ H 3 D Σ exp Σ π M commute, where the vertical arrows are the covering projections (given by quotienting by Γ Σ or Γ). Clearly exp Σ is onto and the restriction of exp Σ to the interior Int(D Σ ) of D Σ is one-to-one and a local isometry. In order to see what the effect of exp Σ is on D Σ, suppose x, y D Σ are interior points of different faces of D Σ for which exp Σ (x) = exp Σ (y). Let Σ 1,..., Σ m lift(σ) be some minimal set of lines so that D Σ = {x H 3 d H 3(x, Σ) d H 3(x, g Σ i ) g Γ Σ i = 1,..., m} (see Lemma 3.1). Then there are unique indices i, j = 1,..., m and unique points x and ỹ in D Σ covering x and y so that x E i and ỹ E j (where E i and E j are (respectively) the equidistant surfaces between Σ and Σ i and between Σ and Σ j ). Now, by the commutivity of the above diagram and since exp Σ (x) = exp Σ (y), π( x) = π(ỹ) and so there is a unique isometry g i Γ so that g i ( x) = ỹ. Since x is in the interior of a face of D Σ, d H 3( x, lift(σ)) = d H 3( x, Σ ) for Σ lift(σ) only if Σ = Σ or Σ i (this uses the fact that the equidistant surfaces E k all meet transversely see Lemma 2.5). Similarly, d H 3(ỹ, lift(σ)) = d H 3(ỹ, Σ ) for Σ lift(σ) only if Σ = Σ or Σ j. Hence g i must take Σ, Σ i to Σ, Σ j (in some order). But since x y, x Γ Σ ỹ and so g i Γ Σ, i.e. g i Σ Σ. Hence g i Σ i = Σ and g i Σ = Σ j. (3.3) We now note that orienting Σ (arbitrarily) defines an orientation on each line in lift(σ), and these orientations are respected by the action of Γ. Then with such an orientations on the lines of lift(σ), the conditions (3.3) determine the isometry g i. Hence g i = g 1 j and each point of E i D Σ is identified with some point of E j D Σ by g i. Hence the faces of D Σ are identified by exp Σ in pairs and so M is the quotient of D Σ by its face-pairings. This observation and the conditions (3.3) are the key to recovering M from the lines Σ, Σ 1,..., Σ m, the group Γ Σ and some combinatorial information. 53

55 We will take up this theme in Section 4.1 and Chapter 6. However, we note at this stage that we can define an invariant of the pair (M, Σ) as follows. Assume that the lines of lift(σ) have been oriented (as above) and that Σ 1,..., Σ m lift(σ) is some minimal collection of lines for which D Σ = {x H 3 d H 3(x, Σ) d H 3(x, g Σ i ) g Γ Σ i = 1,..., m} (such a collection exists by Lemma 3.1). Define d i to be the complex distance (see Definition 1.2) between Σ and Σ i. Note that each line Σ i is well defined up to replacing it by h i Σ i (any h i Γ Σ) and that each d i is unchanged if we reverse the orientations on all lines of lift(σ), so d i is well defined. Then we define the ortholength invariant of the pair (M, Σ) to be (cosh d 1,..., cosh d m ) C m. (3.4) Note that the ordering of d 1,..., d m is not well-defined and that the d i come in pairs, e.g. if there is a face-pairing g i as defined by (3.3) then d i = d j. We will see (in Section 4.1) that the ortholength invariant of a closed hyperbolic 3-manifold (M, Σ), plus some combinatorial data, actually determines (M, Σ) ( cf. Meyerhoff [31]). In Section 3.3 we will generalise the above definitions of D Σ and exp Σ via the following characterisation. We say that a geodesic in M from Σ to a point p M is distance-realising if its length is d M (p, Σ). Then D Σ can be identified with the space of distance-realising geodesics from Σ to the points of M, and under this identification exp Σ becomes the map which sends each geodesic to its end-point. For if γ is a distance-realising geodesic from Σ to a point p M then γ lifts to a geodesic γ which has one end-point on Σ and another end-point p contained in D Σ (since the length of γ equals d M (p, Σ) = d H 3( p, lift(σ))). Quotienting by Γ Σ then gives a well-defined point of D Σ corresponding to γ. Conversely, given a point of D Σ, let p D Σ be a point which covers it and let γ be the geodesic from Σ to p which meets Σ at right-angles. Then the length of γ is d H 3( p, lift(σ)) = d M (p, Σ) (where p = π( p)) and so π γ is a distance-realising geodesic from Σ to p. Note that as a consequence of this identification between D Σ and the space of distance-realising geodesics, exp Σ (Int(D Σ )) M is the set of all points p of M for which there is a unique distance-realising geodesic from Σ to p and that exp Σ ( D Σ ) consists of all points p in M for which there are two or more distance-realising geodesics from Σ to p. 54

56 3.2 Geodesics in a Singular Hyperbolic 3-Manifold In this section we extend some elementary results about geodesics in a Riemannian 3-manifold to the geodesics of compact metric space M which has a smooth hyperbolic structure everywhere except at a set of exceptional points Σ where Dehn surgery-type singularities are allowed (see below). The results of this section follow from elementary theorems on length structures and path metric spaces, and we rely heavily on Gromov [17, 1.A-B] and Bridson-Haefliger [6, I.1, I.3] throughout this section. Recall that a hyperbolic structure on a smooth, orientable 3-manifold is a maximal collection of co-ordinate charts with image in H 3 and whose transition maps are all (restrictions of) orientation-preserving isometries of H 3. A hyperbolic 3-manifold is a smooth 3-manifold which is endowed with a possibly incomplete hyperbolic structure. An isometry between two hyperbolic 3-manifolds is a smooth map from one manifold to the other which is locally an isometry of H 3 when expressed in terms of local co-ordinate charts. If M is a hyperbolic 3-manifold then we can view it uniquely as a Riemannian 3-manifold, so we can define the length l(γ) of a piece-wise smooth path γ : [0, 1] M as l(γ) def = 1 0 γ(t) dt (3.5) where γ(t) is the tangent vector to γ at t [0, 1] and is the norm on each tangent space of M. There is also an induced metric d M on M defined by setting d M (x, y) equal to the infimum of the lengths of all piece-wise smooth paths in M with end-points x, y M. Now, choose a line σ in H 3 and consider the universal cover H 3 of H 3 \ σ, where H 3 is endowed with the unique hyperbolic structure for which the covering projection H 3 H 3 \ σ is an isometry. The Lie group of isometries of H 3 covers the group of isometries of H 3 \ σ. The isometries of H 3 \ σ in the connected component of the identity can be identified with the multiplicative group C of non-zero complex numbers 8. This identification is not quite nat- 8 This identification comes from the fact that the isometries of the upper half-space which fix 0, S 2 each act on S 2 as z az for some a C. 55

57 ural 9, so arbitrarily choose an identification. Then the group of isometries of H 3 in the connected component of the identity can be (uniquely) identified with the additive group C of complex numbers in such a way that the covering projection from the isometries of H 3 to those of H 3 \ σ is the complex exponential map e : C C. Note that under this map, the real axis of C descends to pure translations along σ and the imaginary axis descends to pure rotations of H 3 about σ. In this chapter we study hyperbolic spaces with singularities which are locally modelled on a Dehn surgery model space, or DS model for short, as defined in Thurston [44, 4.4]. Definition 3.2 (DS model) Let L = Z Z be a discrete, co-compact lattice in C. Then L acts on H 3 as a group of orientation-preserving isometries and so the quotient H 3 /L of H 3 by this action has a hyperbolic structure. The completion C of H 3 /L is defined to be a DS model, and the set of added points S = def C \ ( H 3 /L) is the singular set of C. As described in Thurston [44, 4.4] and expanded in Hodgson [22, 2], DS models C come in two types, one with singular set S a circle and the other with S a point. In the first case, C is topologically a manifold and its hyperbolic structure has a cone-type singularity along S (possibly with cone-angle 2π, i.e. no singularity). If S is a point then C isn t even a manifold, however δ-neighbourhoods of S (for any δ > 0) have well-behaved boundaries. We now define the main object of study in this chapter. Definition 3.3 A metric space (M, d M ) is a hyperbolic 3-manifold with Dehn surgery-type singularities along Σ M if: M = M \ Σ is the interior of a smooth, compact, orientable 3-manifold with toroidal boundary components, though in this chapter we restrict to the case of a single boundary component. 9 There are two isometric actions of C on H 3 \ σ, where a C acts as z a z or z (1/a) z (any z S 2 ). 56

58 M has an incomplete hyperbolic structure, and the completion of M equipped with the induced metric is (M, d M ). There is a DS model C with singular set S so that a neighbourhood of Σ in M is isometric (as a metric space) to a neighbourhood of S in C. The focus of this section is on the existence and uniqueness of geodesics in a hyperbolic 3-manifold M which has Dehn surgery-type singularities along a subset Σ. The proofs of these results involve the limits of sequences of paths, and since we are not able to assume a priori that the limits of these sequences are piece-wise smooth, we must now widen our attention to a class of continuous paths in M, namely the rectifiable paths of M. Given any continuous path γ : [0, 1] M, we can form the set of real numbers n { d M (γ(t i ), γ(t i+1 )) 0 = t 0 t 1... t n+1 = 1 for some n}. (3.6) i=0 We say γ is rectifiable if the set (3.6) is bounded. We define the length l(γ) of a rectifiable path γ : [0, 1] M to be the supremum of (3.6). All piecewise smooth curves in M are rectifiable, and it is easy to show that the two definitions of length agree for such curves (see [17, 1.A]). Following Gromov (see [17]) we say that a metric space is a path metric space if the distance between any pair of points is the infimum of the lengths of the (continuous) paths joining them. The following fundamental lemma allows us to apply the results of [17] to hyperbolic 3-manifolds with Dehn surgery-type singularities. Lemma 3.4 Any hyperbolic 3-manifold with Dehn surgery-type singularities is a locally compact path metric space. Note that in the case considered in this chapter, where M has no cusps, this implies that M is compact. Proof of Lemma 3.4: Let M be a hyperbolic 3-manifold which has Dehn surgery-type singularities along Σ. Then by Lemma 3.6 (below) a neighbourhood of Σ is compact. But by the fact that M = M \ σ is the interior 57

59 of a compact 3-manifold (see Definition 3.3) the complement of a slightly smaller open neighbourhood in M is also compact. Hence M is locally compact. Then since (M, d M ) is a path metric space (e.g. see Proposition 3.18 of [6, I.3]), (M, d M ) is also a path metric space by Lemma 3.5 (below). Lemma 3.5 If (X, d) is a path metric space and if its completion (X, d) is locally compact then (X, d) is also a path metric space. Proof: It suffices to prove that for any point x X \ X and any ɛ > 0 there is a path from x to a point of X with length less than ɛ. The lemma then follows by the triangle inequality and the fact that (X, d) is a path metric space. Let x n be a Cauchy sequence in X which approaches x and suppose that d(x 0, x n ) < ɛ/2 for every n > 0. Then for each n > 0 there exists a path c n : [0, 1] X with end-points x 0 and x n with length less than ɛ. We can assume the paths c n are parameterised proportional to arc-length (see Proposition 1.20 of [6, I.1]) and hence that the sequence (c n ) is equicontinuous. Also, each path c n is contained inside the closed ɛ ball about x which, for small enough ɛ, is compact. Hence by the Arzèla-Ascoli Lemma (e.g. see [6, I.3]) there is a convergent subsequence of the paths c n, converging to c, say. Then by the lower semi-continuity of length (see 1.20 and 1.23 of [6, I.1]) the length of c is less than ɛ. Thurston essentially proves the following lemma in [44, 4.4]. Lemma 3.6 If C is a DS model with singular set S then for any δ > 0, the δ-neighbourhood of S in C is compact. Now, we say that a rectifiable path γ : [0, 1] M is parameterised proportional to arc-length if for each t [0, 1], l(γ) t t = l(γ [t, t ]) for all 10 t [0, 1] sufficiently close to t. A continuous path γ : [0, 1] M is locally distance-realising if for each t [0, 1], d M (γ(t), γ(t )) = l(γ [t, t ]) 10 We let [t, t ] denote the compact interval in R bounded by t and t, regardless of whether or not t t. 58

60 for all t [0, 1] sufficiently close to t. Definition 3.7 A geodesic of M is a continuous path γ : [0, 1] M so that for each t [0, 1], d M (γ(t), γ(t )) = l(γ) t t for all t [0, 1] sufficiently close to t. Hence a geodesic is a locally distance-realising path which is parameterised proportional to arc-length. From this it follows that if a geodesic (in the sense of Definition 3.7) has its image inside M then it is a geodesic in the usual differential-geometric sense (e.g. see Corollary 3.9 of Kobayashi-Nomizu [26, Ch. IV]). Now, note that a monotonically reparameterised geodesic is a locally distancerealising path which is not a geodesic. However, the following lemma says that this is the only way such a path can arise. Lemma 3.8 If γ : [0, 1] M is any locally distance-realising path then there is a monotonic function λ : [0, 1] [0, 1] and a unique geodesic γ : [0, 1] M so that γ = γ λ. Proof: See Proposition 1.20 of Bridson-Haefliger [6, I.1]. We now come to the main result of this section, a type of Hopf-Rinow Theorem for M. Proposition 3.9 (Hopf-Rinow) The distance d M (p, q) between any two points p, q M is realised by a geodesic. Proof: By Lemma 3.4, M is a complete, locally compact path metric space so Theorem 1.10 of Gromov [17] applies. Now, from Definition 3.2 it is clear that S is closed in C. Hence by Lemma 3.5, Σ is compact. Hence we have the following corollary to Proposition 3.9. Corollary 3.10 The distance d M (p, Σ) from any p M to Σ is realised by a geodesic. 59

61 3.3 The Tube Domain of a Singular Hyperbolic 3-Manifold In this section we define the tube domain D Σ of a pair (M, Σ), where M is a hyperbolic 3-manifold with Dehn surgery-type singularities along Σ M. We then use the lemmas of the previous section to prove some basic results about D Σ. Given a hyperbolic 3-manifold M with Dehn surgery-type singularities along Σ M (see Definition 3.3) we will define a tube domain D Σ of M based at Σ and a local isometry exp Σ : D Σ M. Let C be the DS model (see Definition 3.2) corresponding to (M, Σ). Then we will think of C as the normal bundle to Σ in M and exp Σ as the corresponding exponential map. Then D Σ C is the set of points of C which correspond to distance-realising geodesics from Σ to the points of M. To make this heuristic precise, first consider some v C. Then there is a unique geodesic γ v : [0, 1] C with end-points on the singular set S and v which realises the distance d C (v, S). This geodesic is a geodesic in the usual differential-geometric sense inside C \ S and γ v is perpendicular to S in the sense that it is perpendicular to the boundary of any δ-neighbourhood of S (for δ > 0 sufficiently small). Now, for any v C define v def = d C (v, S) and note that the geodesic γ v extends to a half-infinite geodesic ray in C. For any t 0 define t v C to be the point on the half-infinite ray though v for which t v = t v. Then in this notation, γ v is the map given by t t v for t [0, 1]. Now, by Definition 3.3 there is a δ > 0 so that the δ-neighbourhood N 1 C of S is isometric to the δ-neighbourhood of Σ in M. The isometry between these two δ-neighbourhoods is not unique but we can (arbitrarily) choose one of them and then define exp Σ to be equal to this isometry in N 1. Then for any v in C \ N 1, there is a well-defined geodesic segment γ v : [0, δ/ v ] M def given by γ v = exp Σ γ v. Since γ v (δ/ v ) M (assuming v = 0) then by classical Riemannian geometry 11 there is a unique extension of γ v to a (linearly reparameterised) geodesic γ v : [0, ɛ 1 ) M for some ɛ 1 > δ/ v. 11 e.g. See Theorem 13 of Spivak [43, vol. 1, Ch. 9]. 60

62 We can define γ v (ɛ 1 ) M by continuity, and if γ v (ɛ 1 ) M then we can again extend γ v to a larger domain. Hence there is some ɛ > δ/ v (possibly ɛ = ) so that γ v : [0, ɛ] M is a geodesic, γ v ((0, ɛ)) M and either ɛ = or else γ v (ɛ) Σ. If ɛ > 1 then we define exp Σ v def = γ v (1) and if ɛ 1 then we say that exp Σ is not defined at v. Let C = {v C exp Σ is defined at v}. If v C then we restrict γ v to a map [0, 1] M. Then we have v = l(γ v ) and γ v = exp Σ γ v. Note also that N 1 C. We now make the following definition (cf. the definition of Dirichlet domain given in [9, 3.5-6]). Definition 3.11 (Tube Domain) The tube domain D Σ of M based at Σ is def D Σ = {v C v = d M (Σ, exp Σ v)}. Hence v C belongs to D Σ if and only if exp Σ is defined at v and γ v : [0, 1] M is a distance-realising geodesic from Σ to its end-point exp Σ v. Conversely, note that if γ : [0, 1] M is a distance-realising geodesic from Σ to x M then there is some v D Σ so that γ v = γ. To see this, note that by classical differential geometry it is enough to prove that some initial segments of γ and γ v coincide. So without loss of generality we can assume that x lies in some δ-neighbourhood of Σ which is globally isometric (via exp Σ ) to a δ-neighbourhood of S. Let v C be such that exp Σ v = x and let γ be the path in C for which γ = exp Σ γ. Then γ is a distance-realising path from S to v so γ = γ v and so γ = γ v as required. Hence there is a one-to-one correspondence between D Σ and the set of distancerealising paths from Σ to the points of M. We will next prove a few elementary facts about the map exp Σ : C M. These proofs use the notion of a developing map, which in rough terms is a coherent piecing-together of the local co-ordinate charts of M. Let M be the universal cover of M endowed with the unique hyperbolic structure which makes the covering projection π : M M a local isometry. 61

63 Then since M is simply connected there exists a local isometry φ M : M H 3 and since M is connected, φ M is unique up to composing on the left by an isometry of H 3 (see [46, 3.4]). This map φ M is the developing map of M. It determines the hyperbolic structure on M in the sense that a map f : U M from a simply connected, open subset U H 3 to M is a local isometry if and only if φ M f : U H 3 is (the restriction of) an isometry of H 3, where f : U M is any lift of f. Lemma 3.12 The set C on which exp Σ is defined is an open subset of C and exp Σ : C M is a local isometry. The fact that exp Σ is a local isometry implies that exp Σ has no conjugate points. Proof of Lemma 3.12: Let B C be any ball in C which is isometric to a ball in H 3, define c B C \ S by c B = v B γ v ((0, 1]) and let c B = c B C. Then c B is contractible so it is simply-connected. Define f 1 : c B H3 by f 1 = φ M ẽxp Σ where ẽxp Σ is some lift of exp Σ c B to a map c B M. Also, define f 2 : c B H3 by lifting c B to H 3 via the covering H 3 C \ S and then applying H 3 H 3 \ σ. Then f 2 is a local co-ordinate chart for the hyperbolic structure on c B and so exp Σ is a local isometry if and only if f 1 = g f 2 for some isometry g of H 3. Now, by the definition of exp Σ, f 1 and f 2 agree up to isometry on c B N 1, where N 1 C is some δ-neighbourhood of S. Hence by altering φ M by an isometry if need be, we can assume f 1 = f 2 on c B N 1. Now, if v c B then f 1 γ v and f 2 γ v are both geodesics 12 in H 3 which agree as maps on (0, δ/ v ]. Hence f 1 γ v = f 2 γ v on [0, 1] and so f 1 = f 2 on c B. So exp Σ is a local isometry at any interior point of c B C. Now, let w C. Since γ w ([δ/ w, 1]) is compact and disjoint from Σ, there is some d > 0 so that the closed 2d-neighbourhood of γ w ([δ/ w, 1]) is contained in M. Assume that d is small enough so that f 2 : c B H 3 is a global (i.e. 12 Recall that γ v : [0, 1] C is a distance-realising geodesic from S to v. 62

64 metric space) isometry on any ball of radius 2d about a point of γ w ([δ/ w, 1]) (such a d exists because f 2 is a local isometry and γ w ([δ/ w, 1]) is compact). Let B be a ball about w of small enough radius that B is embedded and c B lies within a d-neighbourhood of γ w ([0, 1]). Then f 2 (c B ) H 3 lies within a d-neighbourhood N d of f 2 (γ w ([δ/ w, 1])) and so f 1 (c B ) lies in N d as well. But if v c B is such that γ v ((0, 1)) M and γ v (1) Σ then d H 3(f 2 (v), γ w ([δ/ w, 1])) > 2d because f 2 is a global isometry on balls of radius 2d about the points of γ w ([δ/ w, 1]) and because a 2d-neighbourhood of γ w ([δ/ w, 1]) is disjoint from Σ. But v is in the closure of f 1 (c B ) which lies in N d, so this is absurd. Hence such a v cannot exist and so exp Σ must be defined at all points of c B, i.e c B C. Since w C was arbitrary, this proves that C is open. We can now prove the following basic lemma. Lemma 3.13 The tube domain D Σ is compact. Proof: Since M is compact it has finite diameter, and hence D Σ is contained in a δ-neighbourhood of S for some δ > 0. Hence by Lemma 3.6 it is sufficient to prove that D Σ is closed. So let v n D Σ be a sequence of points which converge to some v C. If exp Σ is defined at v then by Lemma 3.12 it is continuous there so v = d M (Σ, exp Σ v ) (see Definition 3.11) and hence v D Σ. So we now must show that exp Σ is defined at v. If exp Σ is not defined at v then let s (0, 1] be the supremum of all t [0, 1] for which exp Σ is defined at t v. Note that exp Σ is defined at s v n for any n > 0. Also, the limit of exp Σ (s v n ) lies in Σ. (This follows by the definition of the domain on which exp Σ is defined.) But s v n D Σ since otherwise there exists a shorter path from σ to exp Σ v n than γ vn, even though γ vn is distance-realising. Also, s v n is bounded above zero since s 0. This implies that the distance from exp Σ (s v n ) to S is bounded above zero, contradicting the fact that the points exp Σ (s v n ) approach a point of Σ. We now have the following useful corollary. 63

65 Corollary 3.14 There is some ɛ > 0 so that exp Σ is defined at all points of the ɛ-neighbourhood of D Σ. Proof: By Lemma 3.12, for each x D Σ there is some r x > 0 so that exp Σ is defined in the ball B(x; r x ) of radius r x about x. The balls B(x; r x /2) of radius r x /2 about points x D Σ cover D Σ and so by Lemma 3.13 a finite number of them also cover D Σ. Then set ɛ equal to half the minimum of the radii of this finite collection of balls. We finish this section with a lemma which will give us certain finiteness results in in Section 3.4. Lemma 3.15 For any point x of M there is some δ > 0 so that there are only finitely many points v C for which exp Σ is defined, exp Σ v = x and v d M (Σ, x) + δ. For δ sufficiently small, each such v corresponds to a distance-realising geodesic γ v from Σ to x. Proof: Suppose the lemma is false, and that there is an infinite sequence of points v i C so that v i converges to d M (Σ, x) and exp Σ v i = x for every i. Then since the set {u C d M (Σ, x) u d M (Σ, x) + 1} is compact, this sequence has a subsequence which is convergent to a point v C for which v = d M (Σ, x). So in any neighbourhood of v there are two or more points which map to x under exp Σ. Since exp Σ is defined at v (by Corollary 3.14) this contradicts the fact that exp Σ is locally injective (see Lemma 3.12). 3.4 Properties of Tube Domains In this section we will use the technical lemmas of the previous sections to prove Theorem 3.16, which gives a detailed description of the tube domain of a hyperbolic 3-manifold with Dehn surgery-type singularities. We state Theorem 3.16 now though we will not formally define the faces of a tube domain until after Lemma 3.17, below. (However, the faces of D Σ 64

66 are certain connected regions in the frontier of D Σ and are analogous to the faces of a Dirichlet polyhedron.) Let M be a hyperbolic 3-manifold with Dehn surgery-type singularities along Σ M (see Definition 3.3) and let D Σ be its corresponding tube domain (see Definition 3.11). Theorem 3.16 To the pair (M, Σ) there is a canonical tube domain D Σ and a local isometry exp Σ : D Σ M which have the following properties. The interior of D Σ is either homeomorphic to a solid torus (if Σ is a circle) or homeomorphic to a solid torus with an unknotted core circle collapsed to a point (if Σ is a point). The boundary D Σ of D Σ has a natural decomposition into a finite number of faces (see below). The map exp Σ : D Σ M is onto and identifies the faces of D Σ in pairs but is a homeomorphism when restricted to the interior of D Σ. The quotient of D Σ by these face-pairings is isometric to M. The set exp Σ ( D Σ ) M consists of all of the points x M for which there are two or more distance-realising geodesics from x to Σ. The proof of Theorem 3.16 is broken into Lemmas 3.17, 3.18 and 3.19 (below) and Lemmas 3.12 and 3.13) (above). From now on we let Q Σ denote exp Σ ( D Σ ) M. Lemma 3.17 The map exp Σ : D Σ M is onto and Q Σ is the set of all points x M for which there are two or more distance-realising geodesics from Σ to x. Also, the restriction of exp Σ to the interior of D Σ is a homeomorphism onto the open and dense set M \ Q Σ. Note that as a consequence of the first part of this lemma, M \ Q Σ consists of all points x M for which there is a unique distance-realising geodesic from x to Σ. Proof of Lemma 3.17: By Corollary 3.10, exp Σ : D Σ M is onto (see also the discussion following Definition 3.11). This implies that the image of the interior Int(D Σ ) of D Σ under exp Σ is dense, because the closure of exp Σ (Int(D Σ )) clearly contains exp Σ (D Σ ) = M. 65

67 For any v D Σ, we will prove that v Int(D Σ ) if and only if there is a unique distance-realising geodesic between Σ and exp Σ v M. This will complete the proof because it implies that the restriction exp Σ Int(D Σ ) of exp Σ to Int(D Σ ) is one-to-one. But since exp Σ is a local isometry (see Lemma 3.12) it maps open balls to open balls. Hence exp Σ (Int(D Σ )) is open and the inverse of exp Σ Int(D Σ ) is continuous. So given v Int(D Σ ), we will show that γ v is the unique shortest geodesic from Σ to exp Σ v. Let u = (1 + ɛ) v C, where ɛ > 0 is small enough so that u D Σ. (Recall from Section 3.3 that (1+ɛ) v is the unique point a distance (1 + ɛ) v away from S on the half-infinite ray in C which contains v). Then γ u is composed of two subgeodesics, γ v and a geodesic arc η with end-points exp Σ v and exp Σ u. Now, suppose (in order to derive a contradiction) that γ is another distance-realising geodesic from Σ to exp Σ v. Then concatenating γ with η gives a path from Σ to exp Σ u which realises the distance d M (exp Σ u, Σ) (assuming ɛ is small). But this path has a corner, so a shorter path from Σ to exp Σ u exists contradiction. Conversely, suppose that γ v is the only distance-realising geodesic from Σ to x def = exp Σ v, where v D Σ is given. We will show that v Int(D Σ ). Let B = B(v; δ) be the ball of radius δ > 0 about v. We assume that δ is small enough so that: B is isometric to a standard ball of radius δ in H 3 ; exp Σ is defined on B and the restriction of exp Σ to B is a global (i.e. metric space) isometry (see Lemma 3.12); δ < ɛ with ɛ as in Corollary 3.14; and the only geodesic from Σ to x of length less than d M (x, Σ) + δ is γ v (see Lemma 3.15). Then the ball B(v; δ/2) of radius δ/2 about v is contained in D Σ. For suppose otherwise, i.e. suppose that there is some w B(v; δ/2) so that w > d M (Σ, y) where y def = exp Σ w. Then since exp Σ (D Σ ) = M, there exists some w D Σ so that exp Σ w = y. But by our assumptions on δ, exp Σ is defined on the ball B(w ; δ/2) of radius δ/2 about w. Also the restriction of exp Σ to B(w ; δ/2) is a global isometry onto the ball of radius δ/2 about y. Hence there is a point v B(w ; δ/2) so that y = exp Σ v. Therefore l(γ v ) < l(γ w ) + δ/2 since d M (x, y) < δ/2 = d M (Σ, y) + δ/2 since w D Σ < l(γ w ) + δ/2 < l(γ v ) + δ since d M (x, y) < δ/2. But by our assumptions on δ, l(γ v ) < l(γ v ) + δ implies v = v. Hence 66

68 w, w B, which contradicts the fact that the restriction of exp Σ to B is one-to-one. Recall that M = M \ Σ and that π : M M denotes the universal covering projection. Let N M be a closed δ-neighbourhood of Σ and let N = N \Σ. We assume that δ > 0 is chosen small enough so that N is disjoint from Q Σ. Now, consider some p Q Σ M. By Lemma 3.17 there are two or more geodesics from Σ to p of length d M (Σ, p). Consider the set of all points p Q Σ for which there are exactly two such geodesics and let ˆF be one connected component of this set. The closure of ˆF is a face of QΣ. By Lemma 3.17, the pre-image of ˆF under expσ : D Σ M is contained in D Σ. There are two connected subsets of D Σ which lie in the pre-image of ˆF under expσ : D Σ M. The closure of each connected component of this pre-image is a face of D Σ. (Note that this definition agrees with the one given in Section 3.1.) Each connected component of the intersection of two faces of D Σ (other than those components which consist of individual points) is an edge of D Σ. Each point in the intersection of three or more faces of D Σ is a vertex of D Σ. The images of the edges and vertices of D Σ under exp Σ are the edges and vertices (respectively) of Q Σ. Lemma 3.18 To any point v 0 D Σ there is some r > 0 and some lines l 0,..., l n in H 3 so that the intersection of D Σ with the ball B(v 0 ; r) of radius r about v 0 is isometric to B(x 0, r) {x H 3 d H 3(x, l 0 ) d H 3(x, l i ) i = 1,..., n} (3.7) where B(x 0, r) is the ball of radius r about some point x 0 H 3 which satisfies d H 3(l 0, x 0 ) = d H 3(l i, x 0 ) for every i = 1,..., n. Also, n is one less than the number of distance-realising geodesics in M from Σ to exp Σ v 0. Each point of the set (3.7) which is equidistant from l 0 and exactly one of the lines l 1,..., l n corresponds to a point in the interior of a face of D Σ. Hence D Σ has only a finite number of faces and the union of these faces contains D Σ. def Proof: Recall that S is the singular core of D Σ, and define D Σ = D Σ \ S and let D Σ be the universal cover of D Σ. We suppose that D Σ is embedded in M 67

69 in such a way that the diagram D Σ M exp Σ M D Σ commutes (where the vertical arrows are covering projections). Since N and Q Σ are disjoint, there is a (unique) connected component Ñ of π 1 (N) contained in the interior of D Σ. Now, consider some v 0 D Σ, and choose any ṽ 0 D Σ M which covers v 0. By Lemma 3.15 there are only a finite number of other points v 1,..., v n of D Σ so that exp Σ v i = p (for any i = 1,..., n) where p = exp Σ v 0. (Note that by the above commuting diagram, π(ṽ 0 ) = p). Let γ vi : [0, 1] M be the corresponding geodesics in M from Σ to p. Let γ i : [0, 1] M be the geodesic from N to p obtained by removing the part of γ vi which is contained in the interior of N. Note that γ i meets N at right-angles and that each γ i has length d M (Σ, p) δ. For each i = 0,..., n there is a unique lift γ i : [0, 1] M of γ i so that γ i(1) = ṽ 0. Let Ñi be the connected component of π 1 (N) which γ i(0) belongs to. Note that Ñ0 = Ñ. Now, recall from Section 3.3 that the developing map of M is a local isometry φ M : M H 3 (and that φ M is unique up to composing on the left with an isometry of H 3 ). From Definition 3.2 it is easy to see that φ M (Ñi) is a δ- neighbourhood of a unique line l i in H 3. Also, since φ M is a local isometry, φ M γ i is a geodesic segment in H 3 which meets the δ-neighbourhood of l i at right-angles and has length d M (Σ, p) δ. Hence d H 3(l i, x 0 ) = d M (Σ, p) for every i = 0,..., n, where x 0 = φ M (ṽ 0 ). Now, choose r > 0 small enough so that φ M is a global isometry from the ball B(ṽ 0 ; r) of radius r about ṽ 0 to the ball B(x 0 ; r), and also assume r < δ 1 /2 where δ 1 is such that if v C and l(γ v ) < d M (Σ, p) + δ 1 then v = v i for some i = 0,..., n (see Lemma 3.15). Then φ M is an isometry from B(ṽ 0 ; r) D Σ to the set (3.7). For suppose that ũ 0 B(ṽ 0 ; r) D Σ and let y 0 = φ M (ũ 0 ) B(x 0 ; r). For each i = 0,..., 1 let u i be the point in B(v i ; r) for which exp Σ u i = q where q def = π(ũ 0 ). Note that u 0 D Σ since ũ 0 D Σ. As for the γ vi above, truncate 68

70 each γ ui by the removing the interior of N, lift this geodesic segment to a path in M which has one end-point at ũ 0, then apply φ M to this lifted path. This gives a geodesic segment from the δ-neighbourhood of l i to y 0. This path meets the δ-neighbourhood of l i perpendicularly and so its length is d H 3(l i, y 0 ) δ. So for each i = 0,..., n, l(γ ui ) = d H 3(l i, y 0 ) (3.8) Since u 0 D Σ, this implies d H 3(l 0, y 0 ) d H 3(l i, y 0 ) for each i = 1,..., n, and so y 0 belongs to (3.7). Conversely, suppose that ũ 0 B(ṽ 0 ; r) and y 0 = φ M (ũ 0 ) B(x 0 ; r) but that ũ 0 D Σ. Then there exists some u D Σ so that exp Σ u = q and l(γ u ) = d M (Σ, q) < l(γ u0 ), where u 0 is as in the previous paragraph (i.e. u 0 is the image of ũ 0 under D Σ D Σ ). Now, since d M (p, q) < r, there is some v B(u; r) so that exp Σ v = p. But l(γ v ) < l(γ u ) + r = d M (Σ, q) + r < d M (Σ, p) + 2r < d M (Σ, p) + δ 1. Hence v = v i for some i = 1,..., n and so u = u i (with u i as in the previous paragraph). So by (3.8) and the fact that l(γ u ) = d M (Σ, q) < l(γ u0 ), y 0 does not belong to the set (3.7). Since the covering projection D Σ D Σ is an isometry, this establishes that B(v; r) D Σ is isometric to the set (3.7). From the above it follows that the faces of D Σ correspond to those points of (3.7) which are equidistant from l 0 and one of the lines l i. Hence the faces of D Σ cover D Σ and by Lemma 2.5 they are also locally finite. Since D Σ is compact (see Lemma 3.13) this implies that the number of faces of D Σ is finite. Lemma 3.19 The tube domain D Σ is either homeomorphic to a solid torus (if Σ is a circle) or homeomorphic to a solid torus with an unknotted core circle collapsed to a point (if Σ is a point). Proof: If v is any point of D Σ then t v Int(D Σ ) for any t [0, 1). (Recall from Section 3.3 that t v is the unique point a distance t v away from S on the half-infinite ray in C which contains v). For suppose that t v Int(D Σ ) for some t < 1. Then by Lemma 3.17, there exists a geodesic γ from Σ to exp Σ (t v) for which l(γ) l(γ t v ). But then concatenating γ with the segment of γ v between exp Σ (t v) and exp Σ v gives a path whose length is less 69

71 than or equal to d M (exp Σ v, Σ). Since this path has a corner at exp Σ (t v), a strictly shorter path exists, which is a contradiction. Now, define a function r : C \ S R by and let v sup{ t v t 0 and t v D Σ } U def = {u C u 1}. Note that the supremum in the definition of r is always achieved (since D Σ is closed) so for any u U \ S, r(u) u D Σ if u = 1 and r(u) u Int(D Σ ) if u < 1. By Lemma 3.18, D Σ is broken into regions consisting of parts of equidistant surfaces, and so r is continuous (in fact, it is piece-wise smooth). Hence there is a homeomorphism U D Σ given by u u if u S and u r(u) u if u U \ S. Since C \ S is the quotient of H 3 by the action of a co-compact lattice L C (see Definition 3.2) and since this action preserves the distance to S, U \ S is homeomorphic to a solid torus minus an unknotted core circle. Then the lemma follows from the comments following Definition The Canonical Dual Ideal Triangulation In this section we will prove that there is a naturally occurring spine Q Σ M (the geometric spine ) corresponding to each pair (M, Σ), where M is a hyperbolic 3-manifold with Dehn surgery-type singularities along Σ M. If M is a hyperbolic 3-manifold (i.e. if the hyperbolic structure is actually smooth at Σ) then we prove that generically Q Σ is a standard spine and hence that there is a canonical ideal triangulation of M \ Σ dual to Q Σ. A general reference for the definitions and results about spines used in this section is [30, 2], but see also [2, E.5-iii]. Note that in [30], standard spines are referred to as special spines. Also note that some of the definitions given below have been restricted to special cases in order to avoid technicalities which are not relevant to our situation. 70

72 Let X be a 3-manifold with non-empty boundary which has a triangulation K X and let Y X correspond to a 2-dimensional sub-complex K Y of K X (in the sense that under some homeomorphism from K X to X, K Y is taken to Y ). Usually Y is said to be a spine of X if X collapses to Y, where the notion of a collapse is defined in terms of the triangulations K X and K Y. However, we will use the following equivalent definition (see [30]) which occurs more naturally in our context. Definition 3.20 Y is a spine of X if X is homeomorphic to the mapping cylinder 13 of some map X Y. Now, as before let M be a hyperbolic 3-manifold with Dehn surgery-type singularities along Σ M and let D Σ and exp Σ : D Σ M be the tube domain and face-pairing projection of the pair (M, Σ) (see Definition 3.11). Then we define the geometric spine 14 def Q Σ of (M, Σ) to be Q Σ = exp Σ ( D Σ ) (as in Section 3.4). By Theorem 3.16, Q Σ consists of those points x M for which there are two or more distance-realising geodesics from Σ to x. Let δ > 0 be small enough so that the closed δ-neighbourhood N δ C of S is contained in the interior of D Σ. Then exp Σ (N δ ) is a closed δ-neighbourhood of Σ in M. Define M δ to be M \ Int(exp Σ (N δ )) where Int(exp Σ (N δ )) is the open δ-neighbourhood of Σ in M. Then M δ is a compact 3-manifold with boundary M δ a 2-torus. Lemma 3.21 The geometric spine Q Σ is a spine for M δ. Proof: Essentially the proof is that M δ is the mapping cylinder of the radial projection of M δ onto Q Σ. First note that there exists a triangulation of M δ so that Q Σ corresponds to a sub-complex. This is because the union of the edges of D Σ is a smooth graph and the interior of each face of D Σ is a smooth 2-manifold, by Lemma Hence the same is true of the edges and faces of Q Σ and so there is a triangulation of Q Σ (see [34]). Define D δ Σ D Σ by D δ Σ = {v D Σ v δ} 13 Recall that the mapping cylinder of a map f : X Y is the quotient of X [0, 1] Y by the relation (x, 1) f(x) for any x X. 14 This terminology anticipates Lemma

73 and note that exp Σ restricts to exp Σ : D δ Σ M δ. Now, the triangulation of Q Σ lifts to a triangulation of D Σ and there is an extension of this to a triangulation of D δ Σ by Theorem XVIII.3.C of [3]. Then since exp Σ is a homeomorphism except on D Σ (see Lemma 3.17) a barycentric subdivision of the triangulation of D δ Σ descends to a triangulation of M δ containing Q Σ as a sub-complex. Now, let r : C \ S R be as in the proof of Lemma 3.19, i.e. r(v) = sup{ t v t 0 and t v D Σ }. Define Φ : DΣ δ [0, 1] M δ by (v, t) exp Σ ((1 t + t r(v)/δ) v). Then by Lemma 3.17, Φ is onto and the restriction of Φ to DΣ δ \ D Σ is a homeomorphism onto M δ \ Q Σ. Define a relation on D Σ by x y if Φ(x) = Φ(y). Then Φ descends to a homeomorphism from the quotient of DΣ δ by to M δ. Hence M δ is homeomorphic to the mapping cylinder of the map N δ Q Σ given by v exp Σ (r(v)/δ v) for any v N δ. Since exp Σ restricts to a homeomorphism from N δ to M δ, this completes the proof. As above, let X be a 3-manifold with non-empty boundary and let Y be a spine of X. Each point of Y which has a neighbourhood (in Y ) homeomorphic to R 2 is called a regular point of Y. The regular points of Y form an open and dense subset of Y (for example, the interior points of the 2-simplices of any triangulation of Y are all regular, though other regular points usually also exist). The points of Y which are not regular are said to be singular. Definition 3.22 A spine Y of X is standard if each connected component of the set of regular points of Y is an open 2-cell and if each singular point of Y has a neighbourhood homeomorphic to either a cone on the graph consisting of a circle and one of its diameters or else to a cone on the 1-skeleton of a 3-simplex (see Figure 3.1). The following lemma guarantees that the spine Q Σ has the local properties of a standard spine, given certain generic conditions. These conditions are generic in the sense that given a generic choice of lines l 0,..., l n as in Lemma 3.18, the conditions of Lemma 3.23 will be satisfied. 72

74 Figure 3.1: The two types of neighbourhoods of singular points in a standard spine Lemma 3.23 Let M be a hyperbolic 3-manifold with Dehn surgery-type singularities along Σ. Suppose that each vertex of D Σ lies in exactly three faces and that exactly three edges of D Σ lie above each edge of Q Σ under the map D Σ M. Then each singular point of the spine Q Σ has a neighbourhood homeomorphic to one of the neighbourhoods illustrated in Figure 3.1. Proof: Let S be a small sphere in M about one of the vertices of Q Σ. Then by Lemmas 3.18 and 2.5, Q Σ determines a smooth, polygonal cell-decomposition on S. The conditions on D Σ in the statement of the lemma imply that the cells of this decomposition are 2-simplices and that exactly three 2-simplices meet at each vertex of the decomposition. A trivial Euler characteristic argument then shows that this decomposition is the standard decomposition of a 2-sphere into four triangles, as in the boundary of a 3-simplex. Recall that M is a (smooth) hyperbolic 3-manifold if the hyperbolic structure on M = M \ Σ extends to a (non-singular) hyperbolic structure on M. Lemma 3.24 Suppose that M is a smooth hyperbolic 3-manifold and that Σ is a simple closed geodesic in M. Then the interiors of the faces of D Σ are open 2-cells. We do not know whether Lemma 3.24 still holds if we allow M to have Dehn surgery-type singularities. 73

75 Proof of Lemma 3.24: We begin by briefly recalling some notation from Section 3.1. If M and Σ are as in the statement of the lemma then we can identify M with the quotient of H 3 by a discrete group Γ PSL 2 C. The pre-image of Σ under the corresponding covering projection H 3 M is a collection lift(σ) of disjoint lines in H 3. We choose some line Σ lift(σ) and then define D Σ H 3 by (3.1). The stabiliser Γ Σ Γ of Σ acts on D Σ and the tube domain D Σ of the pair (M, Σ) can be identified with the quotient of D Σ by Γ Σ. Now, the interior Int(F ) (in D Σ) of a face F of D Σ is by definition one of the connected components of the interior (in D Σ) of E D Σ, where E is the equidistant surface between Σ and some other line Σ lift(σ). The covering D Σ D Σ embeds Int(F ) into D Σ and each face of D Σ is the image of some face F under this covering. Now, by Proposition 2.3, E is diffeomorphic to R 2 and by Lemma 3.18, Int(F ) is a smooth 2-manifold and F \ Int(F ) is a finite collection of piecewise smooth arcs. If the frontier F \ Int(F ) of F is an embedded circle then Int(F ) E is an open 2-cell. If the frontier of F is more complicated then there must exist a hole in F, in the sense that there exists a topological circle c F and a point p E so that p F but p is inside c, i.e. p is contained in the disc component of E \ c instead of the annular component. Now, E D Σ may consist of several faces so it is possible for p F while p E D Σ. But since E D Σ is closed and E \ F is open, without loss of generality we can assume that p E D Σ. Now, E D Σ = {x E d H 3(x, Σ) d H 3(x, Σ )}. (3.9) Σ lift(σ) So if there exists such a circle c and a point p as above then by (3.9) and the definition of the intersection of a collection of sets, there is some Σ lift(σ) so that c H but p H, where H = {x E d H 3(x, Σ) d H 3(x, Σ )}. But this contradicts the fact that H is a union of embedded closed discs in E (from Propositions 2.3 and 2.6). Hence such c and p cannot exist and so F is an embedded disc. We now have the following corollary to Lemmas 3.23 and

76 Theorem 3.25 Let M be a smooth hyperbolic 3-manifold and let Σ be a simple closed geodesic in M. Also assume that each vertex of D Σ lies in exactly three faces and that under the map D Σ M, exactly three edges of D Σ lie above each edge of Q Σ. Then Q Σ is a standard spine of M δ and hence there is a canonical ideal triangulation of M dual to Q Σ. The ideal triangulation of Theorem 3.25 is dual to Q Σ in the sense that inside each tetrahedron, Q Σ has the standard form shown in Figure 3.2. Figure 3.2: The portion of Q Σ contained in one of the tetrahedra of the dual ideal triangulation of M Proof of Theorem 3.25: Note that exp Σ embeds the interior of each face of D Σ into Q Σ, so Lemma 3.24 implies that the interior of each face of Q Σ is an open 2-cell. Since the regular points of Q Σ are exactly the interior points of its faces (by Lemma 3.18), this implies that the connected components of the regular points of Q Σ are open 2-cells. Hence by Lemma 3.23, Q Σ is a standard spine of M δ. From the theory of standard spines (for example, see Theorem 7 of [30]) it follows that there exists a canonical decomposition of M δ into truncated tetrahedra, i.e. tetrahedra minus open neighbourhoods of their vertices. This clearly defines an ideal triangulation of M. 75

77 We note (without proof) that it is fairly clear from Lemmas 3.21 and 3.24 that if M is a smooth hyperbolic 3-manifold then in general there is an ideal cell-decomposition of M dual to Q Σ. One application of Theorem 3.25 is that it gives a general algorithm for computing ideal triangulations for the complements of simple closed geodesics in closed hyperbolic 3-manifolds. This has been automated in the program tube by Goodman-Hodgson [15] and used by Miller [32] to study these complements of simple closed geodesics. Note that Weeks program SnapPea [48] computes ideal triangulations for such complements only for a restricted class of short geodesics. 76

78 Chapter 4 Parameterising Hyperbolic Structures with Ortholengths In this chapter we study the deformation space H(M) of incomplete hyperbolic structures on a 3-manifold 1 M which have Dehn surgery-type singularities when completed. This space was introduced by Thurston to prove his celebrated hyperbolic Dehn surgery theorem (see [44]) and it has been used recently in a proof of the orbifold theorem (see Cooper-Hodgson-Kerckhoff [9]). This deformation space is of continuing interest because there is some hope that major problems in low-dimensional topology such as Thurston s geometrization conjecture may be proved via a thorough understanding of it 2. Our main tool for studying H(M) is an algebraic map Orth K : H(M) C n called the ortholength invariant which locally parameterises H(M). This invariant is defined in terms of an ideal triangulation K of M, but it is actually independent of K in the sense that for sufficiently general K, Orth K determines the ortholength invariant corresponding to any other ideal triangulation (see Corollary 4.8). 1 We also assume that M is orientable and admits a complete, finite-volume hyperbolic structure with a single cusp. 2 For connections between the topology of M and the degeneration of certain types of hyperbolic structures on M, see the survey articles by Kerckhoff [24] and Kojima [28]. 77

79 For a hyperbolic structure whose metric completion is a cone-manifold, the ortholength invariant is essentially cosh of the complex distance from the cone-manifold s singular set to itself along the edges 3 of K. Hence there is a close connection between the ortholength invariant and the tube radius 4. In fact, we will show that under certain conditions, Orth K can be used to calculate the tube radius (see Proposition 4.4 and Theorem 4.13). In recent work of Kojima [27, 5] and Hodgson-Kerckhoff [23] the condition that the tube radius stays bounded away from zero as cone-angles are varied has emerged as a key condition to ensure that the hyperbolic structures do not degenerate. This makes the study of hyperbolic structures via their corresponding ortholengths seem quite promising. We begin this chapter with a discussion of the ortholength invariant in the special case of hyperbolic structures on M which correspond to (topological) Dehn fillings of M (see Section 4.1). We then define the deformation space H(M) (see Section 4.2) before defining the ortholength invariant H(M) C n (see Section 4.3). This definition is given solely in terms of the holonomy representations of H(M). In Section 4.4 we show that Orth K extends to a birational map from the PSL 2 C-character variety X(M) of M into C n. We give an explicit formula for Orth K and then show that Orth K is generically one-to-one on certain interesting components of X(M). In Section 4.5 we prove that as the hyperbolic structure approaches the complete structure, the tube domains of Chapter 3 approach the Ford domain. As an application of this result we give a formula for the tube radius of any incomplete hyperbolic structure in a neighbourhood of the complete structure (see Section 4.6). 4.1 Ortholengths of Smooth Structures In this section we consider the incomplete hyperbolic structures on M which correspond to topological Dehn fillings of M. Given an ideal triangulation K of M we define a corresponding ortholength invariant Orth K and show that the image of this invariant lies on an algebraic variety P(K) C n. We 3 Not all homotopy classes of paths from the singular set to itself contain a distancerealising geodesic, however the ortholength invariant is defined purely in terms of the holonomy representation and so is well-defined in any case. 4 I.e. the radius of the largest embedded tube about the singular set. 78

80 also give a heuristic discussion of the results of Section 4.4 which imply that Orth K is a complete invariant when restricted to the hyperbolic structures considered in this section. Let M be an oriented 3-manifold which admits a complete, finite volume, 1-cusped hyperbolic structure. For example, we could take M to be the figure-8 knot complement and M to be the complete hyperbolic structure on M (see Chapter 5). Hence M is the interior of a compact manifold ˆM whose boundary is a 2-torus (e.g. see Thurston [46, 4.5]). A (topological) Dehn filling of M is the 3-manifold obtained by gluing a solid torus onto ˆM via a diffeomorphism from the boundary of the solid torus to ˆM. The result only depends on the isotopy class of simple closed curves in ˆM which bound disks in the added solid torus. Choosing generators for H 1 ( ˆM; Z) allows us to identify these isotopy classes with pairs of coprime integers. Hence for any pair (p, q) of co-prime integers there is a corresponding Dehn filling of M, which we denote M (p,q). Thurston s Dehn surgery theorem (see [44] and [45]) says that all but a finite number of Dehn fillings M (p,q) of M have hyperbolic structures. From now on we assume that (p, q) is such that M (p,q) admits a hyperbolic structure. But by Mostow rigidity (see Mostow [35] or Gromov-Pansu [18]) this hyperbolic structure is unique, so from now on we take M (p,q) to denote the (p, q)-dehn filling of M endowed with this hyperbolic structure. Now, there is a distinguished simple closed geodesic Σ (p,q) in M (p,q) (corresponding to the core of the solid torus) and (up to isotopy) a diffeomorphism from M to M (p,q) \ Σ (p,q). We can apply the results of Section 3.1 to the hyperbolic 3-manifold M (p,q) and the simple closed geodesic Σ (p,q). If π (p,q) : H 3 M (p,q) is a covering isometry then by Mostow rigidity it is unique up to replacing it by π (p,q) g for some g PSL 2 C. Then as in Section 3.1, if we choose an arbitrary orientation on Σ (p,q) then π 1 (p,q) (Σ (p,q)) is a collection of oriented lines in H 3 which we denote lift(σ (p,q) ). Now, let K be an ideal triangulation 5 of M and denote the edges of K by e 1,..., e n. For each i = 1,..., n, let ẽ i H 3 be some lift of e i under π (p,q). Then since Σ (p,q) is a simple closed curve, for sufficiently small ɛ > 0 the ɛ-neighbourhoods of the lines of lift(σ (p,q) ) will be disjoint. Hence each ẽ i 5 See Definition

81 meets the ɛ-neighbourhoods of exactly two of the lines in lift(σ (p,q) ) (for ɛ small), say Σ and Σ. Then we define the ortholength d i to be the complex distance 6 between Σ and Σ. We define the ortholength invariant Orth K (p, q) of the (p, q)-dehn filling of M to be Orth K (p, q) def = (cosh d 1,..., cosh d n ) C n. (4.1) In Section 4.3 we will prove that Orth K (p, q) is well defined so we will not dwell on this here. Instead, we simply point out that choosing a different lift of e i or a different covering isometry π (p,q) : H 3 M (p,q) has the effect of replacing Σ, Σ by h Σ, h Σ respectively (where h is some orientationpreserving isometry of H 3 ) and this clearly doesn t affect d i. Now, let be one of the tetrahedra of K and let be a lift of to H 3 under π (p,q). For sufficiently small ɛ > 0, the corners of will meet the ɛ-neighbourhoods of exactly four lines Σ 1,..., Σ 4 lift(σ (p,q) ). For i, j = 1,..., 4, let x ij be the hyperbolic cosine of the complex distance between Σ i and Σ j. Then by Proposition 1.11 these complex numbers satisfy the hextet equation 1 x 12 x 13 x 14 0 = det x 21 1 x 23 x 24 x 31 x 32 1 x 34. x 41 x 42 x 43 1 But each edge of covers some edge e k of K, so x ij = cosh d k. Hence to each tetrahedron of K there is a corresponding algebraic equation which is satisfied by the co-ordinates of Orth K (p, q). Hence for each (p, q), Orth K (p, q) lies in an affine algebraic variety P(K). Conversely, since the holonomy representation of a complete hyperbolic 3- manifold determines the manifold (see [46, 3.4]) Proposition 4.6 tells us that Orth K (p, q) determines M (p,q) and hence that Orth K is a complete invariant of the topological Dehn fillings of M. The reason for this is roughly as follows. By the existence and rigidity of hextets (see Proposition 1.11), Orth K (p, q) P(K) determines an (unoriented) hextet corresponding to each tetrahedron 6 The complex distance between two lines in H 3 is the hyperbolic distance between them plus i times an angle of twist see Definition

82 of K (up to isometry). The ideal triangulation K tells us how to fit together copies of these hextets to recover lift(σ (p,q) ). Consider all of the hextets which correspond to a particular tetrahedron of K. There is a unique decktransformation of π (p,q) : H 3 M (p,q) taking one such hextet to another. But conversely, any deck-transformation is determined by the condition that it maps a particular hextet to another. Hence Orth K (p, q) determines the deck-transformations of π (p,q) and so Orth K (p, q) determines M (p,q). In Section 3.1 we defined an ortholength invariant (3.4) of the pair (M (p,q), Σ (p,q) ). For certain ideal triangulations K this invariant agrees with Orth K (p, q). Then since Orth K is a complete invariant on topological Dehn fillings of M, the ortholength invariant of the pair (M (p,q), Σ (p,q) ) plus the combinatorial data embodied by K is also a complete invariant (as claimed in Section 3.1). Meyerhoff has defined the complex ortholength spectrum of a closed hyperbolic 3-manifold as the set of complex distances between pairs of the manifold s simple closed geodesics (see [31]). He has shown that the ortholength spectrum plus some combinatorial data determines the manifold up to isometry. The fact that Orth K is a complete invariant on topological Dehn fillings of M shows that in fact a finite (and computable) subset of the ortholength spectrum, plus slightly stronger combinatorial information than assumed by Meyerhoff, also determines the manifold. 4.2 The Deformation Space of Hyperbolic Structures on M The main object of study in this thesis is the deformation space H(M) of equivalence classes of incomplete, finite volume hyperbolic structures on a 3-manifold M which have Dehn surgery-type singularities when completed. In this section we quickly recall the definition of H(M) but for details the reader is referred to Thurston [46, 3.4] or Goldman [13]. Note that while Mostow-Prasad rigidity (see [35] and [39]) implies that there is a unique complete, finite-volume hyperbolic structure on M, Thurston has shown (see [44] or [45]) that the deformation space H(M) of incomplete structures is at least one complex-dimensional. 81

83 As in Section 4.1 and for the rest of this thesis, let M be an oriented, finite volume, 1-cusped hyperbolic 3-manifold, and let M be the underlying smooth manifold 7. Then M is the interior of a compact 3-manifold ˆM whose boundary ˆM is a 2-torus T 2 (e.g. see Thurston [46, 4.5]). Let ˆN = T 2 [0, 1] be a closed collar neighbourhood of ˆM and let N = M ˆN = T 2 [0, 1). Note that N is a 2-torus. An (oriented) hyperbolic structure on a 3-manifold X is a maximal oriented hyperbolic atlas for X, i.e. a maximal collection of co-ordinate charts from X to H 3 whose transition maps are (the restrictions of) orientation-preserving isometries of H 3. A local diffeomorphism between two hyperbolic manifolds (i.e. manifolds which are each endowed with a hyperbolic structure) is an isometry if it is locally an isometry, i.e. if its expression in terms of local co-ordinates are all restrictions of isometries H 3 H 3. As described in Section 3.2, a hyperbolic structure on M induces a metric on M. In this thesis we will only consider hyperbolic structures which are either complete (with respect to this induced metric) or else are incomplete but whose metric completions are hyperbolic 3-manifolds with Dehn surgerytype singularities (see Definition 3.3). The deformation space H(M) of M is the set of equivalence classes of such hyperbolic structures on M, where two hyperbolic structures on M are said to be equivalent if they are isometric via a diffeomorphism M M which is isotopic to the identity. However, we are primarily interested in the set H(M) H(M) of equivalence classes of incomplete hyperbolic structures on M, so we will also occasionally refer to H(M) as the deformation space of M. In order to define a topology on H(M) we first consider the space PH(M) of pointed hyperbolic structures on M, defined as follows. Choose a base-point M and also choose some M which covers under the universal covering projection π : M M. Then a pointed hyperbolic structure on M is simply a hyperbolic structure plus a choice of isometry ˆf from a small neighbourhood of into H 3 (actually, the germ of such an isometry). As always, these hyperbolic structures are implicitly assumed to either be complete or else to be incomplete but to have Dehn surgery-type singularities when completed. 7 Note that in Chapter 3 M was endowed with a hyperbolic structure, but from now on M will have no such structure. 82

84 Every pointed hyperbolic structure on M has a corresponding developing map f : M H 3 and holonomy representation ρ : π 1 (M, ) PSL 2 C, defined as follows. There is a unique hyperbolic structure on M for which the covering map π : M M is an isometry. With respect to this hyperbolic structure on M, the developing map is the unique local isometry f : M H 3 which equals ˆf π in a neighbourhood of (note that f is a local diffeomorphism). The holonomy representation ρ is determined by f and the equivariance condition that for any α π 1 (M, ), f α = ρ(α) f (4.2) where π 1 (M, ) has been identified with the deck-transformations of π via the choice of base-point. Since the developing map determines the pointed hyperbolic structure, the map from PH(M) to pairs (f, ρ) of developing maps and holonomy representations is one-to-one. Hence we can identify PH(M) with a subset of the space of pairs of local diffeomorphisms M H 3 and representations π 1 (M, ) PSL 2 C. We topologise the local diffeomorphisms with the weak C -topology and the representations with the topology of point-wise convergence. Then PH(M) inherits the subspace topology from this space. (For a more concrete description of this topology, see Lemma 4.1, below.) Now, there is a map PH(M) H(M) given by forgetting each pointed structure s isometry ˆf and then taking equivalence classes of the resulting (unpointed) hyperbolic structure. We give H(M) the quotient topology induced by this map PH(M) H(M). The following lemma says that the topology on PH(M) controls the behaviour of the developing maps away from the ends of M and controls the holonomy representations on any generating set of π 1 (M, ). Lemma 4.1 Suppose we are given real numbers ɛ, ˆɛ > 0, k > 1, a compact set C M, a hyperbolic structure (f t0, ρ t0 ) PH(M) and a finite set of elements α 1,..., α m π 1 (M, ). Then there is a neighbourhood of (f t0, ρ t0 ) in PH(M) so that for any (f t, ρ t ) in this neighbourhood, d H 3(f t (x), f t0 (x)) < ɛ for all points x C. 83

85 If γ : [0, 1] M is a piece-wise smooth path whose image is contained in π 1 (π(c)) then (1/k) l t0 (γ) < l t (γ) < k l t0 (γ) (4.3) where l t (γ) and l t0 (γ) are (respectively) the lengths of γ with respect to the hyperbolic structures (f t, ρ t ) and (f t0, ρ t0 ). For each i = 1,..., m, ρ t (α i ) is ˆɛ-close to ρ t0 (α i ) in PSL 2 C. In the last point of this lemma we take any metric on PSL 2 C which is compatible with its topology. Proof of Lemma 4.1: Given ɛ > 0, k > 1 and a compact set C M as in the statement of the lemma, the weak C -topology on the developing maps guarantees the existence of a neighbourhood of (f t0, ρ t0 ) so that for all (f t, ρ t ) in this neighbourhood and any x C, d H 3(f t (x), f t0 (x)) < ɛ and if u is a unit tangent vector based at x (unit with respect to the (f t0, ρ t0 ) structure on M) then (f t ) (u) (f t0 ) (u) H 3 < 1 1/k. Here H 3 is the norm on the tangent spaces of H 3 and (f t ) : T M T H 3 is the tangent bundle map induced by f t. For any (f t, ρ t ) in this neighbourhood and any tangent vector v based at a point of C, it follows from the above inequalities that (1/k) (f t0 ) (v) H 3 < (f t ) (v) H 3 < k (f t0 ) (v) H 3. (4.4) (This derivation uses the elementary identity k + 1/k > 2.) But by the equivariance (4.2) of f t and f t0, (4.4) is also true for a vector v based at a point of α(c), where α is a deck-transformation of the covering π : M M. In other words, (4.4) is true for any vector v based at a point of π 1 (π(c)). Now, for any smooth path γ : [0, 1] M, recall that l t (γ) = 1 0 (f t ) ( γ t (u)) H 3du where γ t (u) is the tangent vector to γ at parameter value u [0, 1] (see (3.5)). Combining this with (4.4) gives us the second point of the lemma. 84

86 The last point follows immediately from the definition of the topology on the holonomy representations. Define PH(M) PH(M) to be the set of incomplete pointed hyperbolic structures on M and note that the map PH(M) H(M) restricts to a map PH(M) H(M). If we assume, for convenience, that lies in N then for any (f, ρ) PH(M), ρ(π 1 ( N, )) is a non-trivial group of non-parabolic 8, orientation-preserving isometries of H 3 which fixes exactly two points on the sphere at infinity S 2. In fact, it is not hard to see that ρ(π 1 ( N, )) can be identified with the image of the lattice L under the map C C from the isometries of H 3 to those of H 3 (see Definition 3.2 and the discussion preceding Definition 3.2). We finish this section by noting that equivalent hyperbolic structures have conjugate holonomy representations (see [13] or [46, 3.4]). Hence there is a well-defined map from H(M) X(M), where X(M) is the space consisting (roughly speaking) of conjugacy classes of representations π 1 (M, ) PSL 2 C. The space X(M) is called the character variety of M, and we will study this space further in Sections 4.3 and The Ortholength Invariant In this section we define the ortholength invariant corresponding to an ideal triangulation K of M. We show that this invariant is defined at each point of the deformation space H(M) and that it takes values inside an affine algebraic variety P(K) C n. Following Benedetti-Petronio [2, E.5-i] we make the following definition. Definition 4.2 (Ideal Triangulation) A tetrahedral complex consists of a finite number of copies 1,..., n of the standard 3-simplex plus a simplicial isomorphism φ F from each 2-simplex F of 1,..., n to another 2-8 An isometry g PSL 2 C is parabolic if tr 2 g = 4. The group ρ(π 1 ( N, )) cannot contain any parabolics because this would imply that (f, ρ) is complete (see Thurston [44, Ch. 3]). 85

87 simplex of 1,..., n so that if F and F are 2-simplices for which φ F (F ) = F then F F and φ F = φ 1 F. Let denote the equivalence relation on 1... n generated by φ F (x) x for each x F and any 2-simplex F. An ideal triangulation of a 3-manifold M is a tetrahedral complex as above plus a homeomorphism o 1... o n/ M which is smooth on each 3-simplex, where o i denotes i minus its 0-simplices. An infinite ideal triangulation is as above though with the condition that the tetrahedral complex is locally finite instead of the requirement that it has a finite number of simplices. From now on, let K be an ideal triangulation of M with 3-simplices 1,..., n. We call 1,..., n the tetrahedra of K and we call the 1- and 2-simplices of 1,..., n modulo the edges and faces of K (respectively). Corresponding to K, there is an associated infinite ideal triangulation K of the universal cover M of M. The universal cover ˆK of o 1... o n/ can be naturally viewed as an infinite tetrahedral complex. Composing o 1... o n/ M with the covering ˆK o 1... o n/ gives a local homeomorphism ˆK M which lifts to a local homeomorphism ˆK M (by the homotopy lifting property of covering spaces). It is not hard to check that this map ˆK M is actually a homeomorphism and it is clearly smooth on each 3-simplex. Hence ˆK and the homeomorphism ˆK M define an infinite ideal triangulation K of M. Now, let ˆN be a regular neighbourhood of the vertex of 1... n / and let N M be the image of ˆN minus the vertex under the homeomorphism o 1... o n/ M. Then N is a collar neighbourhood of ˆM in M (recall that ˆM is a compact manifold with one toroidal boundary component and that M is the interior of ˆM). Note also that K induces a type of simplicial decomposition of N which is not necessarily a triangulation (e.g. two vertices of a given triangle may coincide) but which is still a nice cell decomposition. We denote this cell-decomposition of N by L. Recall that π : M M denotes the universal cover of M and choose some M which covers the base-point of M (recall also that we assume 86

88 N). Via this choice of, we identify π 1 (M, ) with the deck-transformations of π. Now, let an incomplete pointed hyperbolic structure (f, ρ) PH(M) be given. As noted at the end of Section 4.2, the restrictions imposed on the behaviour of hyperbolic structures (f, ρ) PH(M) in the neighbourhood N imply that ρ(π 1 ( N, )) is a non-trivial group of non-parabolic orientationpreserving isometries of H 3 which fixes exactly two points on the sphere at infinity S 2. Choose an ordering for these two fixed points and let σ denote the oriented line with these (ordered) end-points. Now, since the Euler characteristics of ˆM and ˆM are zero, a simple calculation shows that K has n edges. We let e 1,..., e n denote the edges of K, and for each i = 1,..., n choose some edge ẽ i of K which covers e i. To each end of ẽ i there is a corresponding connected component of π 1 (N). Denote these connected components by Ñ1 and Ñ2 (it is possible that Ñ1 = Ñ2). Also, let Ñ be the connected component of π 1 (N) which contains (hence σ is the core line of f(ñ). Then since Ñ1 and Ñ2 both cover N, there exist deck-transformations γ 1 and γ 2 for which γ i (Ñ) = Ñi (for each i = 1, 2). We define the ortholength d i corresponding to edge e i to be def d i = d C (ρ(γ 1 ) σ, ρ(γ 2 ) σ). While γ 1 and γ 2 are not unique, it is clear that ρ(γ 1 ) σ and ρ(γ 2 ) σ do not depend on their arbitrariness and so ρ(γ 1 ) σ and ρ(γ 2 ) σ are well-defined oriented lines in H 3. Also, the definition of d i doesn t depend on our choice of the edge ẽ i which covers e i. This is because any other edge of K which covers e i is of the form α(ẽ i ) for some deck-transformation α. In the above prescription this has the effect of replacing Ñi by α(ñi), i.e. replacing γ i by αγ i (i = 1, 2). But since d C (ρ(α)ρ(γ 1 ) σ, ρ(α)ρ(γ 2 ) σ) = d C (ρ(γ 1 ) σ, ρ(γ 2 ) σ), d i is not affected by this change. Also, conjugating ρ by some g PSL 2 C has the effect of replacing σ by g σ and each ρ(γ 1 ) by gρ(γ 1 )g 1, which clearly leaves d i unchanged. This shows that d i is independent of our choice of (which we used to identify π 1 (M, ) with the deck-transformations of π) and that d i only depends on the conjugacy class of ρ. Hence the complex valued function cosh d i descends to a function on H(M) (which we also denote cosh d i ). By Lemma 4.1, cosh d i is continuous on PH(M), so cosh d i is also a continuous function on H(M) (by the definition of the quotient topology). We define Orth K : H(M) C n 87

89 to be Orth K ( ) def = (cosh d 1,..., cosh d n ). We think of Orth K as the function which associates cosh of the ortholength d i to edge e i of K for each i = 1,..., n. Now, given a tetrahedron of K, choose some tetrahedron of K which covers it. Each corner of meets a unique connected component of π 1 (N) (which are not necessarily distinct) and these connected components determine four oriented lines ρ(γ 1 ) σ,..., ρ(γ 4 ) σ in H 3. The complex distance between any two of these lines is equal to the ortholength corresponding to one of the edges of. Hence the four oriented lines ρ(γ 1 ) σ,..., ρ(γ 4 ) σ determine a hextet which realises the complex distances associated to the edges of. By Proposition 1.11 this implies that the hyperbolic cosines of these ortholengths satisfy a certain algebraic equation for each tetrahedron of K. Hence the image of Orth K lies in the following (not necessarily irreducible) algebraic variety P(K). Definition 4.3 (The Ortholength Space P(K)) The ortholength space P(K) C n corresponding to the ideal triangulation K with edges e 1,..., e n is the complex affine algebraic variety consisting of those points z = (z 1,..., z n ) C n which satisfy the hextet equations of the tetrahedra of K. If is a tetrahedron of K then its corresponding hextet equation is 1 x 12 x 13 x 14 x 21 1 x 23 x 24 0 = det x 31 x 32 1 x 34 x 41 x 42 x 43 1 = x 2 12x x 2 13x x 2 14x x 13 x 24 x 14 x 23 2x 12 x 34 x 14 x 23 2x 12 x 34 x 13 x 24 where the vertices of have been numbered arbitrarily from 1 to 4 and where we denote cosh of the ortholength associated to the edge of between vertices i and j by x ij = x ji. Note (from the determinant form) that the hextet equation of doesn t depend on the choice of numbering of its vertices. 88

90 Although P(K) lives in an n-dimensional space (one dimension for each edge of K) and satisfies n hextet equations (one for each tetrahedron of K) we will see that for a sensible choice of K, P(K) has an irreducible component which is a complex curve (see Corollary 4.8). See Section 5.1 for a calculation of the ortholength space P(K) when K is the canonical ideal triangulation of the figure-8 knot complement. 4.4 Ortholengths and the Character Variety In this section we extend the definition of the ortholength invariant Orth K given in Section 4.3 to Orth K : X(M) P(K), where X(M) is the PSL 2 C- character variety of M (see below) and the dashed arrow means that Orth K is not necessarily defined at all points of X(M). We show that Orth K is a rational map and that it is generically defined and one-to-one on the irreducible component of X(M) corresponding to the holonomy representations of H(M) (if K is chosen sensibly). This has two important corollaries: The holonomy representation of a hyperbolic structure corresponding to a topological Dehn filling of M determines the hyperbolic structure, so Orth K is a complete invariant of such structures. Since X(M) locally parameterises the deformation space H(M) of M (see [44, 5.2] or [13]) the ortholength invariant also locally parameterises H(M). We also show that if Orth K is defined on the conjugacy class [ρ] X(M) of a representation ρ : π 1 (M, ) PSL 2 C and if Orth K ([ρ]) (1,..., 1) then ρ is irreducible. We begin by briefly sketching the definition and basic properties of the PSL 2 C-character variety and for details and justifications we direct the reader to Boyer-Zhang [4, 3,4]. For related material on the SL 2 C-character variety we refer to Culler-Shalen [10] and the paper [14] by González-Acuña and Montesinos-Amilibia. We do not make the assumption that an algebraic variety is irreducible. The reader may find it convenient to refer to the worked example of Chapter 5 while reading this section. 89

91 The group PSL 2 C can be naturally embedded into SL 3 C as an algebraic subgroup and so the space of representations R(M) of π 1 (M, ) in PSL 2 C is a complex algebraic variety. The PSL 2 C-character variety X(M) of M is the quotient (in the sense of algebraic geometry, see [4, 3,4]) of R(M) by the conjugacy action of PSL 2 C. This space X(M) has a natural algebraic structure which makes it an affine algebraic variety whose co-ordinate ring is the ring of regular functions on R(M) which are invariant under the PSL 2 C- conjugacy action. The ortholength invariant H(M) P(K) C n as defined in Section 4.3 is given solely in terms of holonomy representations. Hence this map H(M) P(K) factors as the composition of the map H(M) X(M), which takes a hyperbolic structure to the conjugacy class of its holonomy representation, and some P(K)-valued function defined on a subset of X(M). Hence it is natural to view the ortholength invariant as a function defined on a subset of X(M). Then the prescription for the ortholength invariant given in Section 4.3 extends without change to the conjugacy class of any representation ρ : π 1 (M, ) PSL 2 C for which ρ(π 1 ( N, )) fixes two points on the sphere at infinity S 2. A simple investigation of the fixed-points of the abelian group ρ(π 1 ( N, )) shows that this condition is equivalent to the requirement that ρ(π 1 ( N, )) is a non-trivial group of non-parabolic isometries and is not isomorphic to Z/2 Z/2 (where Z/2 is the group with two elements). These conditions in turn are equivalent to the algebraic conditions that neither tr 2 ρ(m) = tr 2 ρ(l) = 4 nor tr 2 ρ(m) = tr 2 ρ(l) = tr 2 ρ(ml) = 0, (4.5) where m and l are any pair of generators of π 1 ( N, ) = Z Z. Hence Orth K is defined at all points of X(M) except those which lie in the proper 9 subvariety defined by the equations of (4.5). We write Orth K : X(M) P(K) to indicate the fact that Orth K is not necessarily defined as a set-theoretical function at all points of X(M). We now give a presentation for π 1 (M, ) based on the ideal triangulation K of M. This will be used in Proposition 4.4 (below) to derive an explicit 9 The subvariety is proper since it doesn t contain (the characters of) the holonomy representations of the hyperbolic structures in H(M). 90

92 expression for Orth K. Recall from Section 4.3 that K is an ideal triangulation of M, N is a neighbourhood of the end of M and L is a cell-decomposition of N derived from K. Fix an orientation on the edges of L and on the edges e i of K (i = 1,..., n). Also, choose a maximal tree in L and assume that the basepoint N is a vertex of L. Removing the interior of N from M yields a manifold (with boundary a 2-torus) which is a deformation retract of M. The restriction of the ideal triangulation K to this space is a cell-decomposition W. The 1-cells of W are either 1-cells of L or else come from the edges of K and so the 1-cells of W are oriented. Note also that the maximal tree of L is also a maximal tree for the 1-skeleton of W. Now, there is a standard presentation for π 1 (M, ) corresponding to the celldecomposition W. To each edge of W not contained in the tree there is a generator (represented by a loop contained in the maximal tree except for when it travels along the edge in the direction respecting the edge s orientation) and to each face of W there is a relation. The generators and relations coming from the edges and faces of L reduce to the presentation <m, l ml = lm>. To each edge e i of K there is a corresponding generator α i (i = 1,..., n) and to each face F j of K there is a relator R j (j = 1,..., 2n). Hence we have the following presentation: π 1 (M, ) = <m, l, α 1,..., α n ml = lm, R 1 =... = R 2n = 1>. (4.6) (See Section 5.3 for an example of such a presentation.) The following proposition gives a formula for Orth K in terms of the generators α i of the presentation (4.6). From this formula it is clear that Orth K : X(M) P(K) is a rational map. Proposition 4.4 Let ρ : π 1 (M, ) PSL 2 C be a representation on which Orth K is defined (i.e. for which the conditions (4.5) hold) and for i = 1,..., n let h, g i SL 2 C be matrices which cover ρ(β), ρ(α i ) PSL 2 C, for some β π 1 ( N, ). If tr 2 h 4 then OrthK ([ρ]) P(K) C n has co-ordinates (Orth K ) i given by (Orth K ) i = 2 tr( h g i )tr( h 1 g i ) tr 2 g i tr 2 h

93 for each i = 1,..., n. Note that tr( h g i )tr( h 1 g i ), tr 2 g i and tr 2 h are independent of the choice of matrices h and g i covering ρ(m) and ρ(α i ) and so these three functions define elements of the co-ordinate ring of X(M) (see [4]). Proof of Proposition 4.4: Suppose we have the set-up as in the statement, and let ρ(π 1 ( N, )) fix a geodesic σ of H 3, which we give an arbitrary orientation. Then (Orth K ) i is by definition equal to cosh(d C (σ, g i σ)), where g i = ρ(α i ) PSL 2 C. Since cosh(d C (σ, g i σ)) is invariant under conjugating ρ by an isometry, we can assume h = [ e x/2 0 0 e x/2 ] [ ] a b g i =, c d (where ad bc = 1) for the purposes of calculating cosh(d C (σ, g i σ)). We ll then express our answer in terms which are invariant under conjugacy, and then our formula will hold true for general h and g i. The axis of h has end-points 0 and on the sphere at infinity, and so a half-turn about σ is achieved by [ ] i 0 0 i SL2 C. (There are two matrices covering the half-turn about σ, and our choice here corresponds to choosing an orientation on σ, see Section 1.2.) Then by [ Proposition 1.4 and the fact i 0 1 that a half-turn about g i σ is achieved by g i ] g 0 i i, we have cosh(d C (σ, g i σ)) = 1 [ ] [ ] [ ] [ ] i 0 a b i 0 d b 2 tr 0 i c d 0 i c a = ad + bc = 2ad 1. So our task now is to express ad invariantly. But so tr( h g i ) = e x/2 a + e x/2 d and tr( h 1 g i ) = e x/2 a + e x/2 d tr( h g i )tr( h 1 g i ) = a 2 + d 2 + ad(e x/2 + e x/2 ) = (a + d) 2 2ad + 2ad cosh x = tr 2 g i + 2ad(cosh x 1). 92

94 Combining this with cosh x 1 = 2 cosh 2 (x/2) 2 = (tr 2 h 4)/2 gives us the required formula for (Orth K ) i = cosh(d C (σ, g i σ)). A representation ρ : π 1 (M, ) PSL 2 C is said to be reducible if there is a point on the sphere at infinity S 2 which is fixed by every element of ρ(π 1 (M, )). If ρ : π 1 (M, ) PSL 2 C is a reducible representation on which Orth K ([ρ]) is defined (see (4.5)) then Orth K ([ρ]) = (1,..., 1) P(K) C n. This is because there is a point p S 2 which is fixed by each element of ρ(π 1 (M, )) PSL 2 C, and hence p is an end-point of σ and ρ(α i ) σ for each i = 1,..., n, where σ is the line fixed by ρ(π 1 ( N, )). If we orient σ towards p then ρ(α i ) σ is also oriented towards p, so by Definition 1.2, cosh(d C (σ, ρ(α i ) σ)) = 1 for each i = 1,..., n. On the other hand, Proposition 4.6 (below) says that Orth K is injective when restricted to certain components X K (M) of X(M). Definition 4.5 Define X K (M) to be the union of those irreducible components C of X(M) for which the following holds for any 2-simplex T of K. If a, b, c : X(M) C denote cosh of the ortholengths associated to the edges of T then at a generic point 10 of C, Orth K is defined (see (4.5)) and hence so are a, b and c; a, b and c are not all equal to ±1; and 1 a b 0 det a 1 c. (4.7) b c 1 The condition that a, b and c are not all equal to ±1 ensures that there exists a right-angled hexagon 11 realising a, b and c (see Proposition 1.10). The condition (4.7) implies that this hexagon is non-degenerate (by Lemma 10 We say that a condition holds at a generic point of an irreducible variety V if it holds in the complement of a proper subvariety of V. 11 Recall from the discussion preceding Proposition 1.10 that a right-angled hexagon is determined by three oriented lines in H 3. 93

95 1.13) and this gives us a type of rigidity which is a key technical device for the proof of Proposition 4.6, below. Note that if K contains a tetrahedron which is sufficiently degenerate 12 then the conditions of Definition 4.5 will be violated for every irreducible component C of X(M), and so X K (M) =. However, if K is chosen sensibly then by Lemma 4.7 (below) X K (M) will contain the holonomy representations of H(M). Proposition 4.6 The restriction of Orth K : X(M) P(K) to X K (M) is generically defined and one-to-one on X K (M). We remark that the following proof is constructive. Proof of Proposition 4.6: Let z = Orth K ([ρ]) P(K) where [ρ] X K (M) is a generic point of X K (M) in the sense that the conditions given in Definition 4.5 hold at [ρ] (note that by assumption Orth K is defined there). Let ρ : π 1 (M, ) PSL 2 C be a representative of the conjugacy class [ρ] X(M). We begin by describing some characteristic properties of ρ z before turning to the problem of reconstructing ρ z from z. Let σ be the unique line whose end-points are fixed by ρ(π 1 ( N, )) and suppose that σ is endowed with an (arbitrary) orientation. Recall from Section 4.3 that to each connected component of π 1 (N) there is a corresponding oriented line in H 3 (where π : M M is the covering map and N is essentially a neighbourhood of the cusp of M ) and so each tetrahedron of K determines a hextet 13 H, because each corner of meets a unique connected component of π 1 (N) and so determines a corresponding oriented line. The hextet H realises 14 the ortholengths which Orth K ([ρ]) associates to its edges (more precisely, H realises the ortholengths associated to the edges of the tetrahedron of K which is covered by ). 12 E.g. if an edge of K is homotopically trivial via a homotopy keeping end-points in N. 13 Also recall from the discussion preceding Proposition 1.10 that a hextet is defined by four oriented lines in H This means that to each of the four corners of i there is a corresponding line of H i (though these lines may not all be distinct) and that the complex distances between any pair of these lines is equal to the ortholength associated to the edge joining the corresponding vertices of. 94

96 This association H determines ρ up to conjugacy as follows. Choose a point M covering. This allows us to identify π 1 (M, ) with the decktransformations of π : M M (and this identification is canonical up to composing with an inner automorphism). Then for any tetrahedron of K and any α π 1 (M, ), by the equivariance (4.2) of ρ, α H α = ρ(α) H where ρ(α) H denotes the hextet whose four defining lines are the translates of those of H by ρ(α). Now, by the definition of X K (M), the four lines which form H cannot all coincide, and so the end-points of these lines have at least three points on S 2 (and similarly for H α ). Hence the condition H α = ρ(α) H uniquely determines ρ(α). We will show that z = Orth K ([ρ]) P(K) determines the association H (up to isometry and simultaneous reversal of the orientations of all lines) and hence that z determines [ρ]. To do this we will use two facts: if and cover the same tetrahedron of K then H and H are isometric; and if and share a 2-simplex in K then H and H have three lines in common. Now, by Proposition 1.11, to each tetrahedron i of K (i = 1,..., n), there exists a hextet H i which realises (cosh of) the ortholengths in z = (z 1,..., z n ) P(K) which are associated to the edges of i. Furthermore, H i is unique up to isometry and orientation-reversal (where reversing the orientation of a hextet means simultaneously reversing the orientations of all of its lines we denote H i with reversed orientation by H i ). Let H 1,..., H n (respectively) be some choice of hextets which realise the ortholengths associated to the edges of 1,..., n by z. Any 2-simplex of K is contained in exactly two tetrahedra of K, say i and j. Corresponding to this 2-simplex there are two triplets of lines (which therefore define two right-angled hexagons) contained in H i and H j. These two hexagons both realise the ortholengths of z = (z 1,..., z n ) P(K) which are associated to the edges of the 2-simplex. Hence by Proposition 1.10, after possibly reversing the orientations of H i or H j, these two hexagons will be isometric. We say that the orientations of H i and H j are compatible if the hexagons corresponding to any shared 2-simplices of i and j are isometric (i.e. isometric without first having to reverse the orientations of H i or H j ). 95

97 We say that the orientations of H 1,..., H n are compatible if the orientations of H i and H j are compatible for every i, j = 1,..., n. We have seen above that there exist compatibly oriented hextets which realise the ortholength invariant z = Orth K ([ρ]). But on the other hand, by (4.7) we there are at most two compatible orientations on H 1,..., H n. To see this, note that each of the hexagons contained in any H i (corresponding to a 2- simplex of i ) is non-degenerate (using (4.7) and Lemma 1.13). Hence by Lemma 1.13 this hexagon is distinct (up to isometry) from the corresponding hexagon in H i. This implies that if i and j share a 2-simplex and H i and H j are oriented compatibly then H i and H j are not. So once an orientation on H 1 has been fixed, all of the orientations of compatibly oriented hextets are also fixed. Hence there are two sets of n hextets (up to isometry) which realise the ortholength invariant z and are oriented compatibly. From now on let H 1,..., H n denote one of these compatibly-oriented realisations (note that H 1,..., H n is the other). Now, there is an ideal triangulation K of the universal cover M of M which covers K (see Section 4.3). Each tetrahedron of K covers some i of K, and to we can (arbitrarily) associate a hextet H in H 3 which is isometric to H i. As above, to any 2-simplex of K (contained in and, say) there are two corresponding triplets of lines (which each define a right-angled hexagon) contained in H and H. We say that H and H are compatibly positioned if these two hexagons coincide. We say that an association H from the tetrahedra of K to hextets H in H 3 is compatibly positioned if the two hexagons corresponding to any 2-simplex of K coincide. Consider the two hexagons corresponding to a given 2-simplex of K. Each of these hexagons is defined by a triplet of lines, and since cosh of the ortholengths between them are not all equal to ±1 (see Definition 4.5) each of these triplets of lines has at least four end-points on S 2. This implies that the isometry between these two hexagons is unique. Hence if and share a 2-simplex in K and if H is given then there is at most one H which is compatibly positioned with H. So after fixing an initial hextet (say H 0 corresponding to tetrahedron 0 ) there is at most one way of associating tetrahedra of K to hextets H in such a way that H is compatibly positioned. 96

98 Note that the proof of Proposition 4.6 only used the condition (4.7) to show that up to isometry and simultaneous reversal of orientations there is a unique compatibly oriented set of hextets H 1,..., H n which realise the ortholength invariant z. However, there are less restrictive conditions which guarantee this. As noted earlier, X K (M) may be empty if K contains degenerate tetrahedra. However, if K is chosen sensibly then Lemma 4.7 guarantees that X K (M) will contain the holonomy representation of the complete hyperbolic structure on M. To prove this lemma we recall the following results. Let M (p,q) be the closed hyperbolic manifold obtained by the (topological) (p, q)- Dehn filling of M (assuming (p, q) chosen so that M (p,q) admits a hyperbolic structure) and let Σ (p,q) be the corresponding core geodesic (see Section 4.1). Then the pair (M (p,q), Σ (p,q) ) determines a tube domain (see Section 3.1) and generically there exists an ideal triangulation dual to the geometric spine of (M (p,q), Σ (p,q) ) (see Section 3.5). (Recall that the geometric spine of M (p,q) consists of those points x M (p,q) for which there exists two or more distance-realising geodesics from x to Σ (p,q).) Lemma 4.7 Suppose that K is an ideal triangulation of M which is dual to the geometric spine of some (p, q)-dehn filling of M which admits a hyperbolic structure. Then if p 2 + q 2 is sufficiently large then X K (M) contains the discrete and faithful representation of π 1 (M, ) corresponding to the complete hyperbolic structure on M. In the next section we will show that if Epstein-Penner s ideal cell-decomposition [11] is an ideal triangulation then for p 2 + q 2 sufficiently large, the geometric spine of M (p,q) is dual to this triangulation (see Theorem 4.12). However, note that the assumptions of Lemma 4.7 are weaker than the assumption that Epstein-Penner s canonical cell-decomposition is an ideal triangulation. Proof of Lemma 4.7: Let [ρ (p,q) ] X(M) be the conjugacy class corresponding to the holonomy representation of M (p,q) \ Σ (p,q). (Recall that M (p,q) \ Σ (p,q) is canonically diffeomorphic to M up to isotopy.) As described in Section 4.3, the ortholength invariant Orth K is defined at [ρ (p,q) ] and Orth K ([ρ (p,q) ]) = (cosh d 1,..., cosh d n ) where ±d i C/2πiZ are the corresponding ortholengths. 97

99 The tube radius of M (p,q) about the simple closed geodesic Σ (p,q) is the supremum of the radii r of all r-neighbourhoods of Σ (p,q) which are embedded solid tori. Clearly the tube radius of M (p,q) cannot be bigger than half the real part 15 of any of the ortholengths d i. Fix a covering isometry π (p,q) : H 3 M (p,q) and recall that lift(σ (p,q) ) is the collection of disjoint lines π 1 (p,q) (Σ (p,q)) which define the tube domain of (M (p,q), Σ (p,q) ). If T is a 2-simplex of K then there is an edge e M (p,q) of the spine of M (p,q) which is dual to T. This means that e lies in the intersection of the three faces of the spine which are dual to the edges of T. Hence each point of e has exactly three distance-realising geodesics from Σ (p,q) to itself. For any lift ẽ of e to H 3 there are therefore three lines Σ 1, Σ 2, Σ 3 lift(σ (p,q) ) so that each point of ẽ is equidistant from all three lines Σ i. Also, the complex distance between Σ i and Σ j (for any i, j = 1, 2, 3) will be equal to the ortholength corresponding to one of the edges of T. So denoting cosh of the ortholength invariants of the edges of T by a, b, and c, suppose (in order to derive a contradiction) that 0 = det 1 a b a 1 c. b c 1 Then by Lemma 1.13, the lines Σ 1, Σ 2, Σ 3 form a degenerate hexagon (i.e. they have a common perpendicular). But for large p 2 + q 2, the tube radius of M (p,q) about Σ (p,q) will be large 16 and so the equidistant surface between any Σ i and Σ j will be close to planar (by Proposition 2.3). More precisely, there are two planes which are perpendicular to the common normal to Σ i and Σ j and which sandwich the equidistant surface between them. For arbitrarily large p 2 + q 2 these two planes can be chosen arbitrarily close together. Hence for large enough p 2 + q 2, the equidistant surfaces between Σ 1, Σ 2 and Σ 3 will be disjoint, contradicting the fact that ẽ is contained in their intersection. Hence X K (M) contains the conjugacy class [ρ (p,q) ] of the holonomy representation of the hyperbolic structure M (p,q) \Σ (p,q) on M. Hence X K (M) includes the whole component of X(M) to which [ρ (p,q) ] belongs. For large 15 Recall that the real part of the complex distance between two lines in H 3 is just the usual distance between them, considered as sets of points in H This is part of Thurston s hyperbolic Dehn surgery theorem (see [44]). 98

100 enough p 2 + q 2 this component contains the discrete, faithful representation of π 1 (M, ). A birational equivalence between two algebraic varieties is a rational map from one variety to the other whose image and domain of definition are dense and which has a rational inverse. A lemma of algebraic geometry says that any rational map which is generically one-to-one is a birational equivalence (see Harris [19]). Hence Propositions 4.4 and 4.6 and Lemma 4.7 (plus the comment following Lemma 4.7) have the following corollary. Corollary 4.8 If Epstein-Penner s canonical cell-decomposition of M is an ideal triangulation K then X K (M) contains the discrete, faithful representation of π 1 (M, ). The ortholength invariant Orth K : X(M) P(K) corresponding to K restricts to a birational equivalence between X K (M) and the irreducible components of P(K) which contain the set-theoretical image of X K (M) under Orth K. 4.5 Continuity of Tube Domains By the results of Chapter 3, the metric completion of any incomplete hyperbolic structure in PH(M) has a corresponding tube domain. Also, the complete hyperbolic structure on M has an analogous fundamental domain called the Ford domain (e.g. see [11, 4]). In this section we show that these fundamental domains vary continuously with the points of PH(M). We begin by setting up notation for this section. Recall from Section 4.2 that PH(M) is the space of pointed hyperbolic structures on M which are either complete or are incomplete and have Dehn surgery-type singularities when completed (see Definition 3.3). Each element (f t, ρ t ) PH(M) is specified by its developing map f t : M H 3 and its holonomy representation ρ t : π 1 (M, ) PSL 2 C, where π : M M is the universal cover. The hyperbolic structure (f t, ρ t ) induces a metric d t on M (see the comments following Definition 3.3) and we let M t denote the metric space (M, d t ). The 99

101 hyperbolic structure (f t, ρ t ) on M pulls back to a hyperbolic structure on M. Hopefully without confusion we will also use d t to denote the induced metric on M. If M t is incomplete then its completion M t = M t Σ t is a hyperbolic 3- manifold with Dehn surgery-type singularities along the set Σ t of added points so by Theorem 3.16 there exists a tube domain D t for M t based at Σ t and a local isometry exp Σt : D t M t. The tube domain D t is a hyperbolic solid torus with a possibly-singular core (see Theorem 3.16). Let the complement in D t of this singular core be denoted by D t and let π t : D t M t be the restriction of exp Σt to D t. If M t is complete then there is a canonical Ford domain, which we denote D t, and a surjective local isometry π t : D t M t (see [11, 4]). We will not give a precise definition of the Ford domain here, but we note that π t is a homeomorphism when restricted to the interior of the Ford domain D t and that the image of this restriction consists of those points x M for which there is a unique distance-realising geodesic from x to an embedded horoball neighbourhood of the cusp of M t. For any pointed hyperbolic structure (f t, ρ t ) PH(M) define a tube in M t to be a closed horoball neighbourhood of the cusp of M t if M t is complete and to be a closed, punctured neighbourhood of constant radius about Σ t (i.e. {x M d t (x, Σ t ) δ} for some δ > 0) if M t is incomplete. We say that a tube of M t is embedded if it is disjoint from π t ( D t ). Choose some base-point M, and for each (f t, ρ t ) PH(M) let N t M be the tube of M t for which N t. We assume is chosen so that the tube N t0 of a particular hyperbolic structure (f t0, ρ t0 ) PH(M) is embedded. Also, for any pointed hyperbolic structure (f t, ρ t ) PH(M), any r > 0 and any p M or p M, we let B t (p; r) be the ball (in M or M) of radius r about p with respect to the metric d t. The topology of PH(M) (see Lemma 4.1) gives us very little control over the developing maps on π 1 (N) M, where N M is any set which contains the (non-compact) end of M. However, if N t is embedded for some (f t, ρ t ) PH(M) then the image of π 1 (N t ) under f t is well-behaved: f t (Ñt) is either a closed δ-neighbourhood of some line in H 3 (with δ = d t (, Σ t )) or else a horoball in H 3, for any connected component Ñt of π 1 (N t ). Lemma 4.9 (below) allows us to analyse the behaviour of hyperbolic structures (f t, ρ t ) 100

102 on M by studying the much better behaved geometry of M t \ N t and the geometry of a standard tube which is isometric to N t. This is a standard trick in the theory of deformations of hyperbolic structures (e.g. see [44, 5.8], [27, 5] or [9, 5.4]) so we merely point out that Lemma 4.9 follows from Lemma 4.1 and the following fact from 3-dimensional hyperbolic geometry (cf. [9, 5.4]). For m Z >0, let u m and v m be two sequences of points on S 2 which converge to u and v (respectively) and let p m be a sequence of points in H 3 which converges to some p H 3. If u m v m then let T m be the constant-radius tube about the line with end-points u m and v m for which p m T m and if u m = v m then let T m be the horoball based at u m = v m for which p m T m (for any m Z >0 or m = ). Then inside any ball of H 3, T m approaches T. But this is clear, because in the hyperboloid model (2.1) of H 3, the boundary of T m is given by the equation <x, u m ><x, v m > = <p m, u m ><p m, v m > by Proposition 2.2. Note that if u m = v m then the part of this surface which meets the hyperboloid (2.1) is given by <x, u m > = <p m, u m > and that this is the equation of a horosphere based at u m (e.g. see [11]). Lemma 4.9 For any ɛ 1 > 0 and any (f t0, ρ t0 ) PH(M) there is a neighbourhood U 1 PH(M) of (f t0, ρ t0 ) so that for any (f t, ρ t ) U 1, the tube N t is embedded and N t lies within an ɛ 1 -neighbourhood of N t0 the d t0 -metric). (with respect to From now on we will implicitly assume that any hyperbolic structure (f t, ρ t ) PH(M) is close enough to (f t0, ρ t0 ) so that (for a given small ɛ 1 > 0) the conclusion of Lemma 4.9 holds. Let N i, N o M be the two tubes of M t0 so that N t0 is the ɛ 1 -neighbourhood of N i and N o is the ɛ 1 -neighbourhood of N t0. Then these inner and outer tubes have the property that N i N t N o for any (f t, ρ t ) PH(M) in a neighbourhood of (f t0, ρ t0 ). Roughly speaking, the following lemma says that if we vary the hyperbolic structure (f t0, ρ t0 ) on M by only a small amount then the distance-realising 101

103 geodesics from N t to the points of a small ball in M are homotopic to a finite set of geodesics which only depend on (f t0, ρ t0 ). Lemma 4.10 Let a > 0, p M and (f t0, ρ t0 ) PH(M) be given and suppose that d t0 (π(p), N o ) > a. Then there exists r > 0, a neighbourhood U PH(M) of (f t0, ρ t0 ) and connected components Ñ o (1),..., Ñ o (m) of π 1 (N o ) so that the following holds. Let Ñ (1) t,..., Ñ (m) t be the components of π 1 (N t ) contained in Ñ o (1),..., Ñ o (m) (respectively). Then for any (f t, ρ t ) U, any connected component Ñ t Ñ (1) t,..., Ñ (m) t of π 1 (N t ), and any point x in the ball B t0 (p; r), d t (x, Ñ t) > d t (π(x), N t ). (4.8) The connected components Ñ o (1),..., Ñ o (m) in the lemma are those for which d t0 (x, Ñ (i) t 0 ) = d t0 (π(x), N t0 ). Proof of Lemma 4.10: It follows easily from Lemma 3.15 that there are numbers r, δ > 0 and some connected components Ñ o (1),..., Ñ o (m) of π 1 (N o ) so that d t0 (π(x), N t0 ) + δ < d t0 (x, Ñ t 0 ) (4.9) for any x in the ball B t0 (p; r) and any connected component Ñ t 0 Ñ (1) t 0,..., Ñ (m) t 0 of π 1 (N t0 ). We assume that r is smaller than a, so d t0 (π(x), N o ) > a r > 0. (4.10) Now, for arbitrary k > 1, let U PH(M) be a neighbourhood of (f t0, ρ t0 ) (as in Lemma 4.1) so that (4.3) holds for any (f t, ρ t ) U and any path contained in M \ π 1 (N i ). Let x B t0 (p; r) and let Ñ t Ñ (1) t,..., Ñ (m) t be any connected component of π 1 (N t ). Suppose (in order to derive a contradiction) that d t (x, Ñ t) = d t (π(x), N t ). Then by Proposition 3.9, there exists a d t -distance-realising geodesic γ t from x to Ñ t, i.e. l t (γ t) = d t (π(x), N t ) where l t ( ) is the function which gives the length of a path with respect to (f t, ρ t ). Similarly, if we 102

104 choose j {1,..., m} so that d t0 (x, Ñ (j) t 0 ) = d t0 (π(x), N t0 ) then there is a path γ (j) t 0 from x to Ñ (j) t 0 so that l t0 (γ (j) t 0 ) = d t0 (π(x), N t0 ), where l t0 ( ) gives path-length with respect to (f t0, ρ t0 ). Note that γ t and γ (j) t 0 in M \ π 1 (N i ). Hence in particular (4.3) applies to γ t and γ (j) t 0. are both contained We now use the existence of γ t and γ (j) t 0 and the length estimates (4.3) to estimate d t0 (x, Ñ t 0 ) in terms of d t0 (π(x), N t0 ). The first of the following inequalities comes from the fact that N t lies in an ɛ 1 neighbourhood of N t0 with respect to the hyperbolic structure (f t0, ρ t0 ) (see Lemma 4.9). (1/k) d t0 (x, Ñ t 0 ) (1/k) l t0 (γ t) + 2ɛ 1 /k by Lemma 4.9 < l t (γ t) + 2ɛ 1 /k by (4.3) = d t (x, Ñ t) + 2ɛ 1 /k by the definition of γ t = d t (π(x), N t ) + 2ɛ 1 /k by assumption So multiplying by k gives d t (x, Ñ (j) t ) + 2ɛ 1 /k l t (γ (j) t 0 ) + 2ɛ 1 /k + 2ɛ 1 by Lemma 4.9 < k l t0 (γ (j) t 0 ) + 2ɛ 1 /k + 2ɛ 1 by (4.3) = k d t0 (π(x), N t0 ) + 2ɛ 1 /k + 2ɛ 1 by our choice of j. d t0 (x, Ñ t 0 ) < k 2 d t0 (π(x), N t0 ) + 2ɛ 1 (1 + k). Comparing this with (4.9) and (4.10) yields k 2 1 > δ 2ɛ 1(1 + k). a r By taking U sufficiently small we can assume that k 1 and ɛ 1 are arbitrarily close to zero, giving a contradiction to this inequality. Now, let Ñ o (1),..., Ñ o (m) be given connected components of π 1 (N o ) and for any (f t, ρ t ) PH(M) close to (f t0, ρ t0 ) let Ñ (1) t,..., Ñ (m) t be connected components of π 1 (N t ) which are contained in Ñ o (1),..., Ñ o (m) (respectively). For (f t, ρ t ) PH(M) sufficiently close to (f t0, ρ t0 ) so that (by Lemma 4.9) N t is an embedded tube, each f t (Ñ (i) t ) is either a horoball in H 3 (if (f t, ρ t ) is complete) or else a constant-radius neighbourhood of a line in H 3 (if (f t, ρ t ) 103

105 is incomplete) for each i = 1,..., m. For each j = 2,..., m define the topological half-space H (j) t H 3 to be H (j) t = {x H 3 d H 3(x, f t (Ñ (1) t )) d H 3(x, f t (Ñ (j) t ))}. If f t (Ñ (1) t ) and f t (Ñ (j) t ) are disjoint then the frontier H (j) t of H (j) t is either the equidistant surface between the core lines of f t (Ñ (1) t ) and f t (Ñ (j) t ) (see Section 2.2) or else H (j) t is the equidistant surface between two horoballs (and so is a plane). If f t (Ñ (1) t ) and f t (Ñ (j) t ) intersect then note that H (j) t differs from this description only inside f t (Ñ (1) t ) f t (Ñ (j) t ). Lemma 4.11 Let B be a ball in H 3 which is disjoint from f t (Ñ (i) t ) for each i = 1,..., m and for any (f t, ρ t ) in some neighbourhood of (f t0, ρ t0 ). Then the equidistant surface H (j) t j = 2,..., m. B varies continuously with (f t, ρ t ) for each Since B is disjoint from each f t (Ñ (i) t ), as noted above, H (j) t B is the intersection of B with a surface in H 3 which is equidistant from two lines or two horoballs. Hence in the projective ball model of H 3 (see Section 2.1) this surface is given by a quadratic or a linear equation (see Proposition 2.3). We can topologise the surfaces H (j) t B as elements of the projective space of quadratic surfaces in R 3, i.e. we can topologise the surfaces via the coefficients of their defining equations. It is not hard to check that this topology agrees with the topology induced by the Hausdorff distance in any fixed ball. Proof of Lemma 4.11: A proof of this lemma can be given by studying the formula of Proposition 2.3 as the real part of the complex distance approaches, though this approach is complicated by several technicalities so we will not pursue it here. Instead we follow an algebraic tack which is less conceptual but more rigorous. By Proposition 2.2, the equidistant surface between two lines l 1 and l 2 consists of those points x = (1, x 1, x 2, x 3 ) E 1,3 of the projective ball model (2.2) for which <x, u 1 ><x, v 1 > <u 1, v 1 > = <x, u 2><x, v 2 > <u 2, v 2 > 104 (4.11)

106 where u 1, v 1 are the end-points of l 1 on the sphere at infinity S 2 = {(1, x 1, x 2, x 3 ) E 1,3 x x x 2 3 = 1} and u 2, v 2 are the end-points of l 2 on S 2. Now, Ñ (1) t is fixed by some deck-transformation α 1 of the covering π : M M and so by the equivariance property (4.2) of the holonomy representations ). Let u t 1 and v1 t be the points on S 2 which are fixed by ρ t (α 1 ), with u t 1 = v1 t if there is only one fixed point. As long as ρ t0 (α 1 ) 1 (which we can assume without loss of generality) then u t 1 and v1 t vary continuously with ρ t for (f t, ρ t ) sufficiently close to (f t0, ρ t0 ). If u t 1 v1 t ρ t, ρ t (α 1 ) fixes f t (Ñ (1) t then f t (Ñ (1) t end-points u t 1 and v1. t If u t 1 = v1 t then f t (Ñ (1) ) is a constant-radius neighbourhood of the line l1 t in H 3 with t ) is a horoball based at u t 1. Similarly, there is a deck-transformation α 2 which fixes Ñ (2) t and we let u t 2 and v2 t be the fixed points of ρ t (α 2 ) on S 2. These points vary continuously with ρ t in a neighbourhood of (f t0, ρ t0 ) and they are either the end-points of the core line of f t (Ñ (2) t at which the horoball f t (Ñ (2) t ) (if (f t, ρ t ) is incomplete) or else u t 2 = v2 t is the point ) is based (if (f t, ρ t ) is complete). Combining this with (4.11) shows immediately that if the initial hyperbolic structure (f t0, ρ t0 ) is incomplete then H (2) t B varies continuously with (f t, ρ t ) in a neighbourhood of (f t0, ρ t0 ). Hence from now on we assume that (f t0, ρ t0 ) is complete. There is a deck-transformation α of π : M M which takes Ñ (1) t to Ñ (2) t. Hence by equivariance (4.2), ρ t (α) takes f t (Ñ (1) t ) to f t (Ñ (2) t ). If g t is the Lorentz transformation which acts on the hyperboloid model (2.1) as the isometry ρ t (α) then g t takes u t 1 to a multiple of u2 t and v1 t to a multiple of v2, t say g t u t 1 = a t u t 2 and g t v1 t = b t v2 t for some a t, b t > 0 which vary continuously with (f t, ρ t ). Hence (4.11) is equivalent to <x, u t 1><x, v t 1> = a t b t <x, u t 2><x, v t 2>. (4.12) Note that this equation makes sense even when u t 1 = v1 t and u t 2 = v2 t (i.e. even when (f t, ρ t ) is a complete pointed structure close to (f t0, ρ t0 )) and that its coefficients vary continuously with the hyperbolic structure (f t, ρ t ). Hence the lemma will be proved if we can show that the surface (4.12) is the equidistant surface between f t (Ñ (1) t ) and f t (Ñ (2) t ) for any (f t, ρ t ) in a 105

107 neighbourhood of (f t0, ρ t0 ). As we have already noted, this is true if u t 1 v t 1 and u t 2 v t 2, i.e. if (f t, ρ t ) is incomplete. So we must simply prove it in the case when (f t, ρ t ) is complete. In this case, u t 1 = v t 1 and u t 2 = v t 2 and so the part of the surface (4.12) lying inside the projective ball model (2.2) is given by the equation 0 = <x, u t 1 u t 2 at b t >. (4.13) This is the equation of a plane which is perpendicular to the line with endpoints u t 1 and u t 2. (To see this, take u t 1 = (1, 1, 0, 0) and u t 2 = (1, 1, 0, 0).) Hence (4.13) and the equidistant surface between f t (Ñ (1) t ) and f t (Ñ (2) t ) coincide if and only if they have a point in common. For any hyperbolic structure (f s, ρ s ) in a neighbourhood of (f t0, ρ t0 ) we define the mid-point m s H 3 between f s (Ñ s (1) ) and f s (Ñ s (2) ) to be the unique point which is equidistant from f s (Ñ s (1) ) and f s (Ñ s (2) ) and which minimises the distance to either. This point m s lies on the surface of the type (4.13) for any incomplete pointed hyperbolic structure (f s, ρ s ) and it varies continuously with (f s, ρ s ). Then since the incomplete hyperbolic structures are dense in PH(M) (by Mostow-Prasad rigidity [35], [39] the complete structure consists of a single point in H(M)) m s must lie in the surface of the type (4.13) for any hyperbolic structure (f s, ρ s ). By definition m s lies on the equidistant surface between f s (Ñ s (1) ) and f s (Ñ s (2) ), so this completes the proof. The following theorem tell us that the domains D t and face-pairings π t : D t M vary continuously with the hyperbolic structure (f t, ρ t ) PH(M). Note that this includes our claim that the tube domains of incomplete hyperbolic structures approach the Ford domain of M as the structures become complete. Theorem 4.12 (Continuity of Tube Domains) There is a neighbourhood U PH(M) of (f t0, ρ t0 ) and a finite number of points p 1,..., p k M and positive numbers r 1,..., r k so that for any structure (f t, ρ t ) U, π t ( D t ) k i=1b t (p i ; r i ) and each connected component of πt 1 (B t (p i ; r i )) D t is isometric to B H 3(f t (p); r i ) H (2) t H (m) t (4.14)

108 for some p M and (continuously varying) topological half-spaces H (2) t,..., H (m) t corresponding to some connected components Ñ o (1),..., Ñ o (m) of π 1 (N o ). (Recall that for j = 2,..., m, H (j) t = {x H 3 d H 3(x, f t (Ñ (1) t )) d H 3(x, f t (Ñ (j) t ))} where Ñ (i) t is the connected component of π 1 (N t ) contained in Ñ (i) o.) Proof: The main part of this proof, that each connected component of πt 1 (B t (p; r)) is of the form (4.14) for p π t0 ( D t0 ) and any sufficiently small r > 0, is essentially a corollary of Lemma We now describe in detail how the present theorem follows from this lemma. Let be a base-point for M which covers M under π : M M, and let Ñ t be the connected component of π 1 (N t ) which contains. Consider the universal cover D t of D t to be embedded in M in such a way that D t and the diagram D t M π D t π t commutes, where the vertical arrows are covering projections. Then assuming that N t is embedded (without loss of generality, by Lemma 4.9) we have Ñ t D t and so M D t = {x M d t (x, Ñt) = d t (π(x), N t )}. (4.15) For p D t0, let r > 0 and connected components Ñ (1) t,..., Ñ (m) t of π 1 (N t ) be as in the statement of Lemma Note that by Lemma 4.1, for any (f t, ρ t ) PH(M) sufficiently close to (f t0, ρ t0 ), B t (p; r/2) B t0 (p; r) and so (4.8) holds for any x B t (p; r/2). Assume that r is small enough so that B t (p; r/2) is isometric to a standard hyperbolic ball of radius r/2 for any (f t, ρ t ) close enough to (f t0, ρ t0 ). Hence the restriction of f t to B t (p; r/2) is a global (i.e. metric space) isometry. Now, since p D t0, one of the sets Ñ (1) t,..., Ñ (m) t must be equal to Ñt. By re-labelling we can assume Ñt = Ñ (1) t. Then by (4.15) and Lemma 4.10, D t B t (p; r 2 ) = {x B t(p; r 2 ) d t(x, Ñ (1) t 107 ) d t (x, Ñ (j) t ) j = 2,..., m}. (4.16)

109 Now, for each i = 1,..., m and any (f t, ρ t ) close enough to (f t0, ρ t0 ), d t (x, Ñ (i) t ) is realised by a geodesic γ (i) t. (This follows from Lemma 4.10, application of (4.3) and the fact that d t0 (x, Ñ (i) t 0 ) is realised by a geodesic for small enough r.) Clearly γ (i) t meets Ñ (i) t at right-angles, so f t γ (i) t realises the distance d H 3(f t (p), f t (Ñ (i) t )), and so d t (x, Ñ (i) t ) = d H 3(f t (x), f t (Ñ (i) t )). So by (4.16), f t takes D t B t (p; r/2) to B H 3(f t (p); r/2) H (2) t... H (m) t. Since f t is a metric space isometry on B t (p; r/2), this proves that for any point p D t0 there is a neighbourhood U p of (f t0, ρ t0 ) so that B t (p; r/2) D t is of the form (4.14) for any (f t, ρ t ) U p. Now, by Lemmas 3.13 and 3.17, π t0 ( D t0 ) is compact and so is covered by a finite number of balls B t0 (p i ; r i ) for p i π t0 ( D t0 ) and r i > 0 (i = 1,..., k). Hence by the above work we can assume that each connected component of (B t (p i ; r i )) is of the form (4.14) for any (f t, ρ t ) in some neighbourhood U i of (f t0, ρ t0 ). Hence each component of πt 1 (B t (p i ; r i )) is of the form (4.14) for any (f t, ρ t ) U 1... U k. π 1 t The claim that π t ( D t ) B t (p 1 ; r 1 )... B t (p k ; r k ) for all (f t, ρ t ) sufficiently close to (f t0, ρ t0 ) now follows from Lemma Tube Radii In this section we explain how the continuity of the domains described in Section 4.5 allows us to calculate the tube radii of all incomplete hyperbolic structures on M in a neighbourhood of the complete structure. Specifically, we give a closed form for the square of the cosh of these radii in terms of the ortholength invariant. If (f t, ρ t ) PH(M) is an incomplete hyperbolic structure then we say that a tube in M t is of radius r if it is the r-neighbourhood of Σ t in M t. The tube radius R t of (f t, ρ t ) is defined to be the supremum of the radii of all embedded tubes in M t. Note that by Definition 3.3, R t > 0. Theorem 4.13 If Epstein-Penner s ideal polyhedral decomposition of M (see [11]) is an ideal triangulation K then the tube radius R t of any hyperbolic 108

110 structure (f t, ρ t ) which lies in a neighbourhood of the complete hyperbolic structure is given by R t = 1 2 min{ Re(d 1),..., Re(d n ) } (4.17) where (cosh d 1,..., cosh d n ) is the ortholength invariant corresponding to K and (f t, ρ t ). Any neighbourhood of the complete hyperbolic structure contains all but a finite number of topological Dehn fillings of M (see [44, 5.8]). Hence Theorem 4.13 implies that the formula (4.17) calculates the tube radii of all but a finite number of topological Dehn fillings of M. Also, by Lemma 4.14 (below), the formula (4.17) is equivalent to cosh 2 R t = min( z z 0 1,..., z n z n 1 ) (4.18) where (z 1,..., z n ) = (cosh d 1,..., cosh d n ). Proof of Theorem 4.13: Let (f t0, ρ t0 ) PH(M) be a complete pointed hyperbolic structure and let (f t, ρ t ) PH(M) be a nearby incomplete structure. Also assume that corresponding to the complete structure (f t0, ρ t0 ) on M, Epstein-Penner s ideal polyhedral decomposition of M (see [11]) is an ideal triangulation K. As in Section 4.5, let N t be an embedded tube of M t about Σ t. To any point p in the interior of a face F of the geometric spine π t ( D t ) there is a corresponding (unparameterised) path in M from Σ t to itself obtained by concatenating together the two distance-realising geodesics from Σ t to p. By truncating we obtain a path in M with end-points on N t. Up to homotopy (relative to end-points remaining in N t ) this path doesn t depend on p, so we call it γ F. By Theorem 4.12, the combinatorics of π t ( D t ) are constant. Hence there is an edge e i of K dual to F (see Section 3.5) which is homotopic to γ F, keeping end-points in N t. Let d i denote the ortholength corresponding to e i and (f t, ρ t ) (see Section 4.3). A point m of F is the mid-point of F if it lies in the interior of F and if the two distance-realising geodesics from Σ t to m are tangential. Note that not all faces of π t ( D t ) will necessarily contain a mid-point, however if a mid-point 109

111 m for F does exist then concatenating the two distance-realising geodesics from Σ t to m gives an (unparameterised) geodesic γ with end-points on Σ t. Recall that the real part of the complex distance between two lines in H 3 is (up to sign) just the usual distance between them, considered as sets of points in H 3. Developing γ into H 3 shows that l t (γ) = Re(d i ), where l t (γ) is the length of γ with respect to (f t, ρ t ). Hence R t 1 2 Re(d i). But on the other hand, any tube of radius r < R t in M t is disjoint from π t ( D t ) but the tube T max of radius R t is not. It follows that each point of T max π t ( D t ) is the mid-point of some face of π t ( D t ). Hence R t is the minimum of 1 2 Re(d i) over all edges e i of K which are dual to a face F which contains its mid-point. Let N t0 be a small embedded horoball neighbourhood of the cusp of M t0 and let T be the largest embedded horoball. Let e 1,..., e k be the edges of K dual to the faces of π t0 ( D t0 ) which contain a point of T. Then note that (1) the distance from N t0 to itself along edge e i with i k is strictly less than the distance from N t0 to itself along an edge e j with j > k and (2) each face F t 0 i contains its mid-point if i k, where F t 0 i is the face of π t0 ( D t0 ) dual to edge e i. (Here the mid-point of F t 0 i is the interior point of F t 0 i which minimises the distance to N t0.) By Lemma 4.1 and Theorem 4.12, these two properties are stable in a neighbourhood of (f t0, ρ t0 ). Hence R t is the minimum of 1 Re(d 2 i) over all i = 1,..., n. We note that in general it may be be possible that 1 2 Re(d j) < R t for some edge e j. For instance, this might occur if d j is not represented by a geodesic in M t which has end-points on Σ t and is homotopic to e j in M t. We finish this section with the following useful lemma. Lemma 4.14 If δ, θ R and d = δ + iθ then cosh 2 δ 2 = 1 4 cosh d cosh d Proof: The double-angle formulae for cosh and sinh and the relationship between the hyperbolic and spherical trigonometric functions give and cosh 2 d = cosh 2 δ cos 2 θ + sinh 2 δ sin 2 θ sinh 2 d = sinh 2 δ cos 2 θ + cosh 2 δ sin 2 θ 110

112 and so cosh 2 d + sinh 2 d = 2 cosh 2 δ 1. Substituting d/2 = δ/2 + iθ/2 into the last identity and using the doubleangle formulae again gives the required result. 111

113 Chapter 5 The Figure-8 Knot Complement In this chapter we will consider the results of Chapter 4 in the case when M is the complement in S 3 of the figure-8 knot (see Figure 5.1). In particular, we will calculate the ortholength invariant Orth K : X(M) P(K) and use it to calculate the tube radii of all hyperbolic structures in a neighbourhood of the complete structure. Here we take K to be the canonical ideal triangulation arising from the complete hyperbolic structure on the figure-8 knot complement (see Section 5.1). Figure 5.1: The figure-8 knot 112

114 Proposition 4.4 allows us to express Orth K : X(M) P(K) in terms of (the traces of products of) certain generators of π 1 (M, ) which correspond to the edges of the ideal triangulation K. However, the character variety of the figure-8 knot complement is usually described in terms of (the traces of products of) a different set of generators. Hence we will take a detour (in Sections 5.3 and 5.4) to relate these two set of generators of π 1 (M, ) before calculating Orth K (see Section 5.5) and the tube radius (see Section 5.6). 5.1 The Ortholength Space of the Canonical Triangulation In this section we describe the canonical ideal triangulation K for the figure-8 knot complement and calculate the corresponding ortholength space P(K). The figure-8 knot complement M can be endowed with a complete hyperbolic structure (see [44, 3.1]) and this structure is unique by Mostow-Prasad rigidity (see [35] and [39]). A result of Epstein and Penner (see [11]) shows that every 1-cusped hyperbolic 3-manifold has a canonical ideal cell-decomposition. For the figure-8 knot complement, this cell-decomposition is the ideal triangulation K shown 1 in Figure 5.2 (see [46]). The ideal triangulation K has two edge classes, e 0 and e 1. Let z 0 C be the complex number corresponding to edge class e 0 and let z 1 C correspond to e 1. Then the hextet equation (see Definition 4.3) of tetrahedron 0 is 1 z 0 z 1 z 1 0 = det z 0 1 z 1 z 0 z 1 z 1 1 z 0 z 1 z 0 z 0 1 = 1 3z 2 0 3z z 0 z z 2 0z 1 + z 4 0 2z 3 0z 1 z 2 0z 2 1 2z 0 z z 4 1 = 1 4 (z2 0 + z z 0 z 1 z 0 z 1 1)(2z 1 3z (z 0 1)) (2z 1 3z (z 0 1)). (5.1) 1 The faces of the tetrahedra shown in Figure 5.2 are identified in pairs giving faceclasses A,B,C and D. The class of each face is written at the vertex opposite the face. 113

115 D D e 0 e 1 e 1 e 1 e 0 e 1 A e 0 B C e 1 B e 1 e 0 e 0 e 0 C (a) tetrahedron 0 A (b) tetrahedron 1 Figure 5.2: The canonical ideal triangulation K for the figure-8 knot complement 114

116 The hextet equation corresponding to tetrahedron 1 is identical, so P(K) C 2 is a plane algebraic curve. From (5.1) we see that the defining equation for P(K) splits into three irreducible factors and that P(K) is the union of a conic (whose projective completion is topologically a smooth sphere) and two lines. We will see in Section 5.5 that the quadratic component of P(K) is birationally equivalent to one of the components of the PSL 2 C-character variety of M. It is not clear what significance (if any) the other two components of P(K) have. 5.2 The Character Variety The PSL 2 C-character variety X(M) of M (see Section 4.4) is usually realised as an algebraic curve embedded in C 2, where the co-ordinate functions for this affine space correspond to the generators of the presentation of π 1 (M, ) given in (5.2) below. In order to establish some notation, we will repeat the calculation of this presentation before quoting existing results which give the SL 2 C- and PSL 2 C-character varieties of M. I thank Stephan Tillmann for showing me his calculations of X(M) and the presentation (5.2). Figure 5.3 shows the Wirtinger generators (see Rolfsen [41, 3.D]) for π 1 (M, ) and a base-point for M. Usually the base-point for the Wirtinger presentation would be chosen to be at the vantage point of Figure 5.3. Then each generator would correspond to a loop in M which starts at the vantage point, travels down to the start of the corresponding arrow, along the arrow under the knot and then back up to the vantage point. In order to get loops based at instead of the vantage point of Figure 5.3, we conjugate the usual Wirtinger generators by the path which travels from along the line of sight to the vantage point. This identifies π 1 (M, ) with the group of the following presentation: π 1 (M, ) = <t 1, t 2, t 3, t 4 t 3 = t 1 1 t 2 t 1, t 4 = t 2 t 3 t 1 2, t 2 = t 4 t 1 t 1 4, t 1 = t 1 3 t 4 t 3 >. We can clearly eliminate the generators t 3 and t 4 using the first two relations. 115

117 t 1 t 2 t 3 t 4 * Figure 5.3: The Wirtinger generators for π 1 (M, ) Doing so gives us the 2-generator presentation π 1 (M, ) = <t 1, t 2 [t 1 2, t 1 ]t 1 2 = t 1 1 [t 1 2, t 1 ]> (5.2) where [s, t] def = sts 1 t 1 is the commutator of s and t. We define two complex algebraic functions X and Y on the space of representations of π 1 (M, ) into SL 2 C by X( ρ) = tr( ρ(t 1 )) = tr( ρ(t 2 )) and Y ( ρ) = tr( ρ(t 1 t 2 )) for any representation ρ : π 1 (M, ) SL 2 C. These functions are clearly conjugacy invariant, so they descend to functions (also denoted X and Y ) on the SL 2 C-character variety 2. In fact, X and Y are the co-ordinate functions of an embedding of the SL 2 C-character variety in C 2 as the complex curve given by 0 = (X 2 Y 2)(X 2 Y 2X 2 Y 2 + Y + 1) (5.3) (see González-Acuña and Montesinos-Amilibia s paper [14]). 2 Similarly to the PSL 2 C-character variety, the SL 2 C-character variety of M is defined to be the algebro-geometric quotient of the space of representations π 1 (M, ) SL 2 C by the conjugacy action of SL 2 C as defined in Culler-Shalen [10]. 116

118 Now, from the presentation (5.2) above it is clear that each representation ρ : π 1 (M, ) PSL 2 C lifts to a representation ρ : π 1 (M, ) SL 2 C. Since this representation is only arbitrary up to replacing it by ρ there are welldefined algebraic functions U and V defined on the PSL 2 C-character variety X(M) by U(ρ) = X 2 ( ρ) and V (ρ) = X 2 ( ρ) Y ( ρ). (i.e. U and V are elements of the co-ordinate ring of X(M).) By (5.3), these functions define an embedding of X(M) into C 2 as the complex curve given by the equation 0 = (V 2)(V 2 UV + V + U 1). (5.4) From (5.4) it is easy to see that X(M) is composed of two irreducible components, one being the line V = 2 and the other being the rational curve U = V 2 + V 1. (5.5) V 1 The line V = 2 corresponds to conjugacy classes of reducible 3 representations π 1 (M, ) PSL 2 C and the conic given by (5.5) corresponds to conjugacy classes of irreducible representations (see [14]). Hence the conic component contains the PSL 2 C-conjugacy classes of the two discrete, faithful representations of π 1 (M, ) into PSL 2 C. Both of these representations are holonomy representations of the hyperbolic structure on M. Note that while there are two such representations up to conjugacy by the orientation preserving isometries PSL 2 C of H 3, by Mostow-Prasad rigidity (see [35] and [39]) there is only one conjugacy class if we conjugate by the full group of hyperbolic isometries (i.e. including the orientation-reversing ones). Hence the co-ordinates of the two representations in X(M) C 2 are complex conjugates of each other. Recall that Orth K is defined at the points of [ρ] X(M) where neither tr 2 ρ(m) = tr 2 ρ(l) = 4 nor tr 2 ρ(m) = tr 2 ρ(l) = tr 2 ρ(ml) = 0, where m and l are any pair of generators for π 1 ( N, ). Then since t 2 is one of a pair of generators for π 1 ( N, ) (see Figure 5.3) and U = tr 2 ρ(t 2 ), we see immediately from (5.4) that Orth K is defined everywhere on X(M) except perhaps at at a finite number of points. 3 A representation ρ : π 1 (M, ) PSL 2 C is reducible if there is a point on the sphere at infinity S 2 which is fixed by every element of ρ(π 1 (M, )). 117

119 In Appendix A we calculate tr 2 ρ(l), where l π 1 ( N, ) and t 2 (as above) generate π 1 ( N, ). On the line V = 2, tr 2 ρ(l) = 4 (by (A.14)) and on the conic component of X(M) we have tr 2 ρ(l) = (U 2 5U + 2) 2 (by (A.15)). From this we can calculate that there are no points of X(M) where tr 2 ρ(t 2 ) = tr 2 ρ(l) = tr 2 ρ(t 2 l) = 0 and that the points of X(M) where tr 2 ρ(t 2 ) = tr 2 ρ(l) = 4 are exactly those points (u, v) X(M) for which u = 4. On the conic (5.5) there are two points which lie on the line U = 4, namely (4, 1 ± ζ) where ζ C is a non-trivial cube-root of 1. These points correspond to two PSL 2 C-conjugacy classes of the discrete, faithful representations of π 1 (M, ) described above. The only point of the line V = 2 which also lies on the line U = 4 is obviously the point (4, 2). This point corresponds to the trivial representation of π 1 (M, ). 5.3 The Presentation of π 1 (M, ) based on K In this section we will write down the presentation (4.6) for the canonical ideal triangulation K of M (see Section 5.1). Our interest in this presentation comes from the fact that the formula for Orth K in Proposition 4.4 is given in terms of its generators. Following the notation of Chapter 4 we take N M to be a closed tubular neighbourhood of the figure-8 knot minus the knot itself. If we truncate the ideal tetrahedra of K by removing the interior of N (see Figure 5.4) then we obtain a cell-decomposition L of the boundary N of N (see Figure 5.5). We choose the maximal tree for the 1-skeleton of L which is shown in bold in Figure 5.6. We also choose a base-point in N and two generators m and l for π 1 ( N, ) as shown in Figure 5.6. We will see (in Figure 5.7) that this base-point corresponds to the same point in S 3 as the base-point chosen in Section 5.2, so this choice of matches the previous choice. Then the presentation (4.6) for π 1 (M, ) becomes π 1 (M, ) = <α 0, α 1, m, l α 0 = [m 1, α 1 ] (5.6) 118 α 1 = [α 0, m] (5.7) l = α0 1 [α1 1, m 1 ] (5.8) ml = lm>

120 n 1 n 10 e 0 n 2 n 0 e 1 e 1 n 8 n 11 e 0 n 4 n 5 e 0 e 1 n 11 n 0 e 1 n n7 4 n 3 e 1 n 8 n 6 n 7 n 10 e 0 n 9 n 3 n 5 e 0 e 1 n2 n 1 n 9 e 0 n 6 (a) tetrahedron 0 (b) tetrahedron 1 Figure 5.4: The tetrahedra of K, truncated to show the edges of the corresponding cell-decomposition L of N n 2 n 11 n 7 n 3 n0 n 8 n0 n n 1 6 n 10 n 5 n 9 n n n n 4 n 3 Figure 5.5: The cell-decomposition L of N, viewed from outside of M 119

121 * m l Figure 5.6: The maximal tree of L and the generators for π 1 ( N, ) where [s, t] = sts 1 t 1 is the commutator of s and t. The generator α 0 is the unique loop (up to homotopy) based at which remains in the maximal tree except when it travels along the edge e 0 (in the direction in which e 0 is oriented). Hence α 0 is represented by the loop composed of edges n 1 11, e 0 and n 7 (in that order), where n 1 11 denotes the path which traverses the edge n 11 in the direction counter to the orientation of n 11. Similarly, α 1 is represented by the loop composed of edges e 1, n 1 2 and n 11, in that order. The first three relations (5.6), (5.7) and (5.8) of the presentation based on K correspond to faces A, B and C of K. The relation corresponding to face D can be expressed as a product of conjugates of these relations and so is superfluous. It is not hard to see that in general, the relations of (4.6) corresponding to three of the hexagonal faces of a truncated tetrahedron imply the relation corresponding to the fourth hexagonal face (assuming the generators m and l of π 1 ( N, ) commute). 120

122 5.4 The Natural Isomorphism between the Two Presentations The presentation for π 1 (M, ) based on K (see Section 5.3) and the presentation (5.2) are not just abstract presentations for π 1 (M, ). By construction, their generators correspond to particular homotopy classes of loops in M, so there is a natural way to identify the abstract group of each presentation with π 1 (M, ). Hence there is a natural isomorphism between the two presentations and the goal of the present section is to calculate this isomorphism. The first step is to find the loops in M S 3 which correspond to the generators of each presentation. We already know these for the presentation (5.2) (see Figure 5.3) so we now need to find the loops which correspond to the generators of the presentation based on K. This is an easy task given Figure 5.7, and the results are shown in Figure 5.8. I thank Sally Miller for showing me the embedding of the ideal triangulation K into S 3 as the figure-8 knot complement which underpins Figure 5.7. e 0 C A e 1 * B D Figure 5.7: The edge classes and the maximal tree in S 3 From Figures 5.8 and 5.3 and using the relations t 3 = t 1 1 t 2 t 1 and t 4 = t 2 t 3 t 1 2 = t 2 t 1 1 t 2 t 1 t 1 2 to eliminate the generators t 3 and t 4 it is easy to calculate the natural isomorphism between the two presentations of π 1 (M, ). 121

123 α 0 * l m * α 1 * (a) The generators for π 1 ( N, ) (b) The generator α 0 (c) The generator α 1 Figure 5.8: The loops corresponding to the generators of the presentation for π 1 (M, ) based on K This isomorphism is given by m t 2, and the inverse isomorphism is given by l t 1 t 1 2 t 1 1 t 2 t 1 t 2 t 1 1 t 1 2 t 1 t 1 2, α 0 t 2 t 1 1 t 2 t 1 t 1 2 t 1 1, α 1 t 1 1 t 2 t 1 t 2 t 1 1 t2 1 t 1 t 1 2 (5.9) t 1 lα 1 1 m and t 2 m. As a check on the isomorphism (5.9), we finish this section by showing that under it the relator of (5.2) maps to the identity. First we have (5.10). α 1 = [α 0, m] by (5.7) = [[m 1, α 1 ], m] by (5.6) = m 1 α 1 mα1 1 mα 1 m 1 α1 1 and so pre-multiplying both sides by m 1 α 1 1 m gives m 1 α 1 1 mα 1 = α 1 1 mα 1 m 1 α 1 1. (5.10) 122

124 Hence under the isomorphism (5.9) [t 1 2, t 1 ]t 1 2 [t 1, t 1 2 ]t 1 m 1 lα 1 1 mα 1 m 1 α 1 1 m 1 α 1 mα 1 1 m = m 1 lm 1 α 1 1 mα 1 m 1 α 1 mα 1 1 m by (5.10) = m 1 lm 1 α 1 1 mα 1 α 0 m by (5.6) = 1 by (5.8) 5.5 The Ortholength Invariant In this section we use Proposition 4.4 to give an explicit formula for the ortholength invariant Orth K : X(M) P(K) corresponding to the canonical ideal triangulation K of the figure-8 knot complement. For any representation ρ : π 1 (M, ) PSL 2 C with conjugacy class [ρ] X(M), the formula for Orth K ([ρ]) in Proposition 4.4 is given in terms of matrices in SL 2 C which cover ρ(m), ρ(α 0 ) and ρ(α 1 ). Instead of choosing these matrices randomly, choose a lift ρ : π 1 (M, ) SL 2 C (this always exists see Section 5.2). For convenience, define tr ρ : π 1 (M, ) C by tr ρ (γ) def = tr( ρ(γ)) for any γ π 1 (M, ). The formula for Orth K ([ρ]) in Proposition 4.4 is therefore given in terms of tr ρ (m), tr ρ (α 0 ), tr ρ (mα 0 ), tr ρ (m 1 α 0 ), tr ρ (α 1 ), tr ρ (mα 1 ) and tr ρ (m 1 α 1 ). We will first express these traces in terms of the functions X = X( ρ) = tr( ρ(t 1 )) = tr( ρ(t 2 )) and Y = Y ( ρ) = tr( ρ(t 1 t 2 )) defined on the SL 2 C-character variety (see Section 5.2). It will then become apparent that we can express Orth K in terms of functions U and V which are defined on the PSL 2 C-character variety X(M). These trace calculations use the isomorphism (5.9), the relations (5.6) (5.8), the conjugacy invariance of the trace of a matrix and the fact that tr(ab) + tr(a 1 B) = tr(a)tr(b) (5.11) for any A, B SL 2 C. (The formula (5.11) follows from the fact that A + A 1 = tr A I for any A SL 2 C, where I is the 2 2 identity matrix.) Since 123

125 these trace calculations are not very illuminating we place them in Appendix A and simply give a summary here: tr ρ (m) = X tr ρ (α 0 ) = X 2 Y X 2 Y 1 tr ρ (mα 0 ) = X tr ρ (m 1 α 0 ) = X X2 Y X 2 Y 1 X tr ρ (α 1 ) = X 2 Y tr ρ (mα 1 ) = X tr ρ (m 1 α 1 ) = X(X 2 Y ) X. (5.12) Substituting the traces (5.12) into the expression for Orth K : X(M) P(K) given in Proposition 4.4 gives, in terms of the functions U = X 2 and V = X 2 Y : UV V 1 (Orth K ) 0 = 2 U ( V V 1 )2 1 U 4 (Orth K ) 1 = 2 UV U V 2 1. (5.13) U 4 Now, as described in Section 5.2, X(M) is composed of two components, V = 2 and U = V 2 +V 1 (see (5.4) and (5.5)) which correspond to reducible V 1 and irreducible representations respectively. Also, Orth K is defined at all points of X(M) except those points which correspond to either trivial or discrete and faithful representations, namely the points (4, 2) and (4, 1 ± ζ), where ζ C is a non-trivial cube-root of 1 (see Section 5.2). By substituting V = 2 into (5.13) we see that Orth K maps the line V = 2 to the point (1, 1) P(K), as expected (see the discussion preceding Definition 4.5). On the conic component of X(M) expression (5.13) reduces to Orth K (u, v) = 1 v 2 3v + 3 ( v2 + 3v 1, v 2 v 1) (5.14) for any (u, v) X(M) C 2. From this it is easy to check that the image of Orth K lies inside the conic component of P(K) by substituting expression 124

126 (5.14) into its defining equation (see (5.1)) and verifying that the resulting expression in v vanishes identically. Now, a quick calculation using Figure 5.2 and the expression (5.14) shows that the conditions (4.7) of Definition 4.5 are 0 (V 2) 2 (V 2 + V 1) and 0 (V 2 3V + 2) 2 (V 2 + V 1). Hence X K (M) is the conic component of X(M) and so by Proposition 4.6 the restriction of Orth K to this component is generically injective. This implies that the map (5.14) has a rational inverse, and a simple calculation shows that this is given by (z 0, z 1 ) (u, v) for any (z 0, z 1 ) on the conic component of the curve (5.1), where v = 2z 0 + z z and u is given by the equation (5.5). 5.6 Tube Radii In this section we use the expression for the ortholength invariant given in Section 5.5 to calculate the tube radii of all hyperbolic structures in H(M) which lie in a neighbourhood of the complete structure. Let R(u, v) denote the tube radius 4 of the figure-8 knot complement M equipped with the hyperbolic structure corresponding to (u, v) X(M). Then by Theorem 4.13 and Lemma 4.14, for any hyperbolic structure in some neighbourhood of the complete structure, cosh 2 R(u, v) = min( z z 0 1, z z 1 1 ) (5.15) where (z 0, z 1 ) = Orth K (u, v) is the ortholength corresponding to the canonical ideal triangulation of M evaluated at the point (u, v) X(M). Substituting the expression (5.14) into (5.15) and simplifying therefore gives: cosh 2 R(u, v) = min(1 + v2 3v + 2, v v 2 ) 2 v 1 u 4 (5.16) 4 See Section

127 where we have used the identity v 2 3v + 3 = (v 1)(u 4), as follows from (5.5). Using (5.5) we can convert (5.16) into an expression only involving u, i.e. only involving the square of the trace of the meridian (this involves a choice of orientation for the complete structure). An assymtotic formula for the tube radius in terms of Dehn surgery co-ordinates can then be derived from the assymtotic expansions of the log-holonomy of the meridian given in Neumann- Zagier [36, 6]. 126

128 Chapter 6 Constructing Hyperbolic Structures via Tube Domains Thurston s Dehn surgery theorem (see [44] or [45]) implies that if M is a 3- manifold which admits a complete, finite-volume hyperbolic structure with a single cusp then there is a one complex-dimensional family of incomplete hyperbolic structures on M which have Dehn surgery-type singularities when completed. Thurston has also given an ingenious method for calculating these structures in terms of an ideal triangulation of M (see Thurston [44] or Neumann-Zagier [36]). This construction (which has been automated by Weeks [48] in a program called SnapPea) attempts to parameterise hyperbolic structures in terms of complex tetrahedral parameters which satisfy certain consistency equations. Whenever the solutions to the consistency equations are tetrahedral parameters with positive imaginary parts then they correspond to a genuine decomposition of M into geometrical parts. However, when the consistency equations have solutions with negative imaginary parts then there is no obvious interpretation of the decomposition of M and Thurston s method fails to give a corresponding hyperbolic structure. This tetrahedral construction attempts to calculate a hyperbolic structure for an arbitrary hyperbolic Dehn filling of M. Since not all Dehn fillings admit a hyperbolic structure 1 it is not so surprising that this construction 1 For example, the trivial filling on a knot complement is S 3 and so cannot sustain a 127

129 sometimes fails! However it is known that there are Dehn fillings which admit hyperbolic structures and for which the tetrahedral construction fails. For example, the cone-manifolds over the figure-8 knot complement (see 5.1) with cone angle strictly less than 2π/3 all have hyperbolic structures (see [21] and also [22, 29]) but for cone angles bigger than 2π/ the tetrahedral construction fails. In fact, Choi [8] has shown that there cannot exist a positively-oriented ideal triangulation for the cone-manifold over the figure-8 knot complement when cone-angles are 2π/ (i.e. when one of the tetrahedra of the canonical ideal triangulation flattens out). In this chapter we present a method for constructing incomplete hyperbolic structures on M which have Dehn surgery-type singularities when completed. Given an ideal triangulation K of M and a point z P 0 (K) (where P 0 (K) is a certain irreducible component of the ortholength space P(K) introduced in Section 4.3) we will construct a hyperbolic tube T z which is homeomorphic to a solid torus minus a core geodesic. Under certain conditions there exists a face-pairing quotient map T z M which identifies points of the boundary of tube in such a way that the hyperbolic structure on T z descends to an incomplete hyperbolic structure on M. If this occurs then T z is the tube domain (see Chapter 3 and Section 4.5) of the corresponding hyperbolic structure on M. This tube construction has been automated by Oliver Goodman in a program [15] called tube. This program has been used to calculate many examples of hyperbolic structures where the tetrahedral construction fails. For instance, tube has calculated a hyperbolic structure for m004(4,0) (i.e. the hyperbolic cone-manifold with cone-angles π/2 along the figure-8 knot) and for m007(3,1) (i.e. the manifold of third lowest volume on SnapPea s closed census). (Note that SnapPea fails to compute a hyperbolic structure for m004(4,0) and that there is no known positively oriented tetrahedral decomposition of m007(3,1).) The tube domains for m004(4,0) and m007(3,1) (and two steps to them from the complete structure) are illustrated in Figures 6.1 and 6.2. Each of these figures shows the universal cover of the torus which links the core geodesic. The regions shown are the faces of the tube domain projected onto the linking torus. In these figures the horizontal direction corresponds to distance along the core geodesic and the vertical direction corresponds to the angle around the geodesic. hyperbolic structure. 128

130 We begin this chapter by defining the tube T z (in Section 6.1) and the facepairings of T z (in Section 6.2). In Section 6.3 we give certain conditions under which quotienting T z by its face-pairings gives an incomplete hyperbolic structure on M. We finish with a discussion of various questions and directions for future work suggested by the results of this thesis (see Section 6.4). The work presented in Sections 6.1 and 6.2 is based on joint work with Oliver Goodman and Craig Hodgson. (a) m004(10,0) (b) m004(5,0) (c) m004(4,0) Figure 6.1: Tube domains for cone-manifolds singular along the figure-8 knot complement 129

131 (a) m007(9,3) (b) m007(6,2) (c) m007(3,1) Figure 6.2: Tube domains for m007(3,1) and some related cone-manifolds 130