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THE ENDING LAMINATION THEOREM 0. Preface This Paper Is THE ENDING LAMINATION THEOREM BRIAN H. BOWDITCH Abstract. We give a proof of the Ending Lamination Theorem, due (in the indecomposable case) to Brock, Canary and Minsky. 0. Preface This paper is based on a combination of two earlier manuscripts both originally produced in 2005. Many of the ideas for the first preprint were worked out while visiting the Max Planck Institute in Bonn. The remainder and most of the write- up of this was carried out when I was visiting the Tokyo Institute of Technology at the invitation of Sadayoshi Kojima. Most of the second preprint was written while visiting the Centre Bernoulli, E.P.F. Lausanne, as part of the programme organised by Christophe Bavard, Peter Buser and Ruth Kellerhals. I thank all three institutions for their generous hospitality. I also thank Dave Gabai for his many helpful comments on the former preprint; in particular, for the argument that appears at the end of Section 2.3, and his permission to include it here. The final drafts of these preprints were prepared at the University of Southampton. The preprints were substantially reworked and combined, and new material added at the University of Warwick. I thank Al Marden for his comments on the first few sections of the current manuscript. I also thank Makoto Sakuma and Eriko Hironaka for their interest and comments. Contents 0. Preface 1 1. General Background 3 1.1. Introduction 3 1.2. An informal summary of the Ending Lamination Theorem 5 1.3. Basic notions 8 1.4. Background to 3-manifolds 11 1.5. End invariants 16 1.6. Ingredients of the proof 23 Date: First draft: September 2011. Revised: 11th April 2020. 1 2 BRIAN H. BOWDITCH 1.7. Outline of the proof of the Ending Lamination Theorem 28 2. The indecomposable case of the Ending Lamination Theorem 32 2.1. Surface groups 32 2.2. The curve graph 35 2.3. Topology of products 39 2.4. Annulus systems 47 2.5. The combinatorial structure of the model 57 2.6. Margulis tubes 63 2.7. Systems of convex sets 68 2.8. The model space 75 2.9. Bounded geometry 82 2.10. Lower bounds 87 2.11. Families of proper subsurfaces 99 2.12. Controlling the map on thick parts 104 2.13. The doubly degenerate case 110 2.14. The proof of the main theorem in the indecomposable case 115 3. The general case of the Ending Lamination Theorem 124 3.1. Compressible ends 125 3.2. Pleating surfaces 127 3.3. Negatively curved spaces 130 3.4. Isolating simply degenerate ends 134 3.5. Quasiprojections and a-priori bounds 138 3.6. Short geodesics in degenerate ends 143 3.7. The model space of a degenerate end 145 3.8. Proof of the Ending Lamination Theorem in the general case 149 4. Promotion to a bilipschitz map 151 4.1. Subsurfaces in triangulated 3-manifolds 152 4.2. A topological surgery construction 157 4.3. Subfibres in product spaces 159 4.4. Constructing a bilipschitz map 164 4.5. The bilipschitz theorem 172 5. Appendix 177 5.1. The uniform injectivity theorem 177 References 183 THE ENDING LAMINATION THEOREM 3 1. General Background 1.1. Introduction. In this paper, we give an account of the Ending Lamination Theorem. This was proven in the \indecomposable" case by Minsky, Brock and Canary [Mi4, BrocCM], who also announced a proof in the general case. It gives a reasonably complete understanding of the geometry of complete hyperbolic 3-manifolds with finitely generated fundamental group. An informal statement is given as Theorem 1.2.1 here, and a more formal statement as Theorem 1.5.4. The Ending Lamination Conjecture originates in the seminal work of Thurston in the late 1970s [Th1]. As well as a plethora of new results, this threw up many new questions (see Section 6 of [Th2]). Undoubtedly the most important of these was the Geometrisation Conjecture which asserts that any compact 3-manifold can canonically be cut into \geometric" pieces | each admitting a metric locally modelled on one of eight geometries (Question 1 of [Th2]). It is hyperbolic ge- ometry that provides the richest source of these examples, so that it might be said, in a certain sense, that \most" 3-manifolds are hyperbolic. Three other key conjectures related to non-compact hyperbolic 3-manifolds, namely the Tame- ness Conjecture, the Ending Lamination Conjecture and the Density Conjecture (respectively, Questions 5, 11 and 6 of [Th2]). Major progress towards these con- jectures as subsequently made by many authors. Then, in the space of three years, between 2002 and 2004, proofs of the first three conjectures were produced: re- spectively by Perelman [Pe], by Agol and Calegari and Gabai [Ag, CalG], and by Minsky, Brock and Canary [Mi4, BrocCM]. This paved the way to a proof of the Density Conjecture, finally completed in [Oh, NS]. As well as its intrinsic interest, the Ending Lamination Theorem, and the technology involved in its proof, has many application beyond 3-manifolds. For example, it has consequences for the geometry of the mapping class groups and Teichm¨ullerspace. In 2 dimensions, it is well known that a closed surface of genus at least 2 admits a rich \Teichm¨ullerspace" of hyperbolic structures: in other words metrics locally modelled on the hyperbolic plane. In contrast, in higher dimensions, a closed manifold admits at most one hyperbolic structure. This is the Mostow Rigidity Theorem [Most]. If we allow non-compact manifolds, then in general, we might expect to get many such structures. In dimensions greater than 3, there are lots of interesting examples; though there is no well developed general theory, and it would seem that hyperbolic structures do not arise in such a natural way. The focus has therefore been on the 3-dimensional situation. To outline this, let M be a complete hyperbolic 3-manifold. We can write 3 3 ∼ M = H =Γ, where H is hyperbolic 3-space, and Γ = π1(M) acts isometrically 4 BRIAN H. BOWDITCH and freely properly discontinuously on H3. In general, one could construct lots of exotic examples, so it is natural to restrict to the case where Γ is finitely generated. In this case the Tameness Theorem tells us that M is homeomorphic to the interior of a compact manifold with (possibly empty) boundary. This is equivalent to saying that each end, e, of M has a neighbourhood homeomorphic to Σ × [0; 1), where Σ = Σ(e) is a closed surface. In general the situation may be complicated by the existence of cusps, where a given curve in M can be made arbitrarily short by homotoping it out one of the ends. To simplify the discussion, we will assume in this introdcution that M has no cusps. In this case, each Σ(e) has genus at least 2. A consequence of (the proof of) the Tameness Theorem is that we can associate to e an \end invariant", a(e). In fact, there are two cases. Maybe e is \geometrically finite" and a(e) is an element of the Teichm¨ullerspace of Σ(e). Or else, e is \degenerate" and a(e) is an \ending lamination". The latter can be loosely thought of as an ideal point of the Teichm¨ullerspace (though that is not entirely accurate). In the case where there are no cusps, the Ending Lamination Theorem tells us that M is determined up to isometry by its topology and its family of end invariants. There is a reasonably good understanding of the structure of end invariants, and so this gives a fairly satisfactory answer to the question of what such 3-manifolds look like. (Though of course, other questions remain open.) In particular, if M is closed, then there are no end invariants. Therefore, the Ending Lamination Theorem implies the Mostow Rigidity Theorem. An important case is where M is homeomorphic to Σ × R. (In this case, Tameness is due to Bonahon [Bon].) One can very loosely think of the geometry of M determining a coarse path in the Teichm¨ullerspace of Σ, parameterised by the R-coordinate. If both ends are degenerate, then this path is bi-infinite, and we can think of it as limiting on the respective ending laminations. Again, this is only very approximately true (though it can be made more precise in certain cases, see [Mi1] for example). In this way, a connection can be made with the geometry of Teichm¨ullerspace. Indeed, understanding this particular case, is in some sense, the key to understanding the Ending Lamination Theorem in general. The proof of the Ending Lamination Theorem here, as in [Mi4, BrocCM], works by constructing a model space. This is a riemannian manifold, P , homeomorphic to M, whose construction depends only on the data given by the (degenerate) end invariants. One then constructs a bilipschitz map from P to M. If M 0 is a hyperbolic 3-manifold homeomorphic to M and has the same end invariants, we therefore get a bilipschitz map from M to M 0, via P . Some standard results from the theory of quasiconformal maps allows us to promote this to an isometry. The construction of P is of intrinsic interest, since it makes much use of the geometry of curve complexes, and so one obtains new information about these spaces: both along the way and in retrospect. The fact that the map from P to M is lipschitz, though highly non-trivial, is significantly simpler than showing that it has a lipschitz inverse. In fact, for THE ENDING LAMINATION THEOREM 5 the proof of the Ending Lamination Theorem, we only need a weaker statement, namely that the map lifts to a quasi-isometry of universal covers.
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