The Ending Lamination Theorem Brian H. Bowditch Mathematics Institute, University of Warwick Coventry CV4 7AL, Great Britain. [Preliminary draft: September 2011] Preface This paper is based on a combination of two earlier manuscripts respectively enti- tled “Geometric models for hyperbolic 3-manifolds” and “End invariants of hyperbolic 3-manifolds”, 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 7, and his permission to include it. The final drafts of these preprints were prepared at the University of Southampton. The preprints were substantially reworked and combined, and some new material added at the University of Warwick. I thank Al Marden for his comments on the first few sections of the current manuscript. 0. Introduction. In this paper, we give an account of Thurston’s Ending Lamination Conjecture regard- ing hyperbolic 3-manifolds. This was originally proven for indecomposable 3-manifolds by Minsky, Brock and Canary [Mi4,BrocCM1], who also announced a proof in the general case [BrocCM2]. Since then, a number of other approaches have been proposed (see for example, [R,BrocBES,So2]). In the present work, we give a proof of the general case of the Ending Lamination Theorem. It is intended to be self-contained, given the background material discussed in Section 2. The Ending Lamination Theorem is a major component of the classification of finitely generated Kleinian groups, or equivalently complete hyperbolic 3-manifolds with finitely generated fundamental group. This project is now essentially complete. The Ending Lami- nation Theorem can be viewed as the “uniqueness” part of the classification. It shows that such manifolds are determined by their “end invariants”. The other main components of the classification are the Tameness Theorem [Bon,Ag,CalG] and the existence of manifolds with prescribed end invariants (see [NS]). We proceed with an informal statement of the Ending Lamination Theorem. For a more precise statement, see Theorem 2.4. Let M be a complete hyperbolic 3-manifold with π1(M) finitely generated. The “thin 1 elc1 part” of M is the (open) subset where the injectivity radius is less than some sufficently small fixed “Margulis” constant. The unbounded components of the thin part form a (possibly empty) finite set of cusps of M. Removing these unbounded components, we obtain the “non-cuspidal” part, Ψ = Ψ(M) of M. The Tameness Theorem tells us that Ψ is topologically finite. This means that there is a compact manifold, Ψ,¯ with boundary ∂Ψ,¯ and a closed subsurface ∂I Ψ¯ ∂Ψ,¯ such that Ψ(M) is homeomorphic to Ψ¯ ∂I Ψ.¯ Fixing some such (proper homotopy⊆ class of) homeomorphism, we can identify ∂Ψ(M\) with ∂V Ψ = ∂Ψ¯ ∂I Ψ,¯ which we refer to as the vertical boundary of Ψ. Each torus component \ of ∂Ψ bounds a Z Z-cusp of M, and does not meet ∂I Ψ. All other components of ∂Ψ have genus at least⊕ 2. Note that the ends of Ψ are in bijective correspondence with the components of ∂I Ψ. Each end e has a neighbourhood homeomorphic to Σ [0, ), where Σ = Σ(e) is such a × ∞ component. Note that this meets ∂V Ψ in ∂Σ [0, ). Associated to each such end we have a geometric “end invariant”, which is either× a Riemann∞ surface (for a “geometrically finite” end) or a geodesic lamination (for a “degenerate” end). The Ending Lamination Theorem asserts: Theorem 0 : M is determined up to isometry by the topology of its non-cuspidal part, Ψ(M), together with its end invariants. Implicit in this is a preferred proper homotopy class of homeomorphism of Ψ with a given topological model. This gives rise to “markings” of the end invariants which we assume to be part of the data. The homotopy class of the isometry of the will respect these markings. One can get a simpler picture by considering the case where M has no cusps, so that Ψ(M) = M. In this case, Ψ¯ is a compactification of M, obtained by adjoining a surface (of genus at least 2) to each end of M. These surfaces are just the components of ∂Ψ = ∂I Ψ,¯ and each has an end invariant associated to it. The above will be dicsussed in more detail in Section 2. Particular cases of the Ending Lamination Theorem were known before the work of Minksy et al. If M is closed, so that Ψ(M) = M and there are no end invariants, then M is determined by its topology. This is Mostow rigidity [Most] in dimension 3. The same applies in the more general case, where M has finite volume, so that each boundary component of Ψ(M) is a torus corresponding to a Z Z-cusp of M. Again, M is determined by its topology. This was shown in [Mar] and [P]. More⊕ generally still, if M is geometrically finite (i.e. all its ends are geometrically finite) then the Ending Lamination Theorem is shown in [Mar]. (This used arguments similar to those of Section 16 here.) Indeed there was a complete classification in this case, following from the deformation theory of Alfors, Bers, Marden, Maskit etc. (see [Mar]). Our account of the Ending Lamination Theorem broadly follows the strategy of the original, though the logic is somewhat different. Notably, we take the “a-priori bounds” theorem of Minsky [Mi4] as a starting point, rather than a result embedded in the proof. An independent argument for this is given in [Bow4]. (See also [So3] for some simplifications of this.) The model spaces we use are essentially the same as those in [Mi4], though we give 2 elc1 a combinatorial description that bypasses much of the theory of heirarchies as developed in [MaM2]. We shall first prove the theorem in the indecomposible case. Some additional ingre- dient will be needed for the general case, mainly to give a proper description of the end invariants, and to “isolate” an end of Ψ from the “core” of the manifold. Apart from that, it will only call for reinterpreting certain constructions. We will present the main part of the argument in the specific context of a doubly degenerate manifold, namely, where Ψ(M) is a topological product, Ψ(M) ∼= Σ R, and both ends are degenerate. We do this for several reasons. × Firstly it greatly simplifies the exposition. Most of the main ideas can be seen in this context. What remains for the general indecomposable case is largely a matter of describing how the various bits fit together in a more complicated situation. Secondly, these ideas have further applications to Teichm¨uller theory and the geometry of the curve complex, etc. As far as these are concerned, one only really needs to worry about such product manifolds. In the case of a doubly degenerate group, we get a somewhat cleaner, and stronger statement. In particular, one can show that the quasi-isometry constants are uniform, in that they depend only on the topology of the base surface. (A similar uniformity in this case is obtained in [BroCM1].) This is lost (at least without more work) in the general indecomposable case. A third, though relatively minor, reason is that one is obliged to give some special consideration to the doubly degenerate case, since there we have to check that each end of the model gets sent to “right” end of M — a fact that is automatic from the topology in all other situations. (Indeed, precisely this issue caused Bonahon a certain amount of strife in [Bon].) In the first five sections of this paper, we describe the relevant bacground, outline the main results, and describe the ingredients of the proof. In Sections 6 to 16, we proceed to a proof of the Ending Lamination Theorem in the indecomposible case. In Sections 17 to 24, we describe the modifications necessary to deal with the general case. In Section 25, we give an account of the Uniform Injectivity Theorem applicable to our situation. 1. Background. We give a summary of the main ideas behind the Ending Lamination Conjecture and the classification of finitely generated kleinian groups. We include some historical background, though our account is not strictly chronological. Much of the discussion of this section is not logically essential to understanding the statement or proof as presented in this paper. The bits that are will be reviewed again later. In particular, a more formal discussion of end invariants will be given in Section 2. Before the late 1970s, much of the theory of 3-manifolds and of kleinian groups had developed separately. Prior to this, most major results of 3-manifold theory were based on combinatorial or topological techniques. General accounts of the topological theory of 3-manifolds can be found in [He] and [J]. Meanwhile, the theory of Kleinian groups tended to use analytical machinery, focusing on the action of the group on the Riemann sphere. 3 elc1 An account of the state of the art with regard to Kleinian groups around this time can be found in [BersK]. In the background, though never fully exploited, was hyperbolic geometry arising from the fact that a kleinian group acts properly discontinuously on hyperbolic 3- space.