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The Ending Laminations Theorem Direct from Teichmüller Geodesics

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The Ending Laminations Theorem Direct from Teichmüller Geodesics The Ending Laminations Theorem direct from Teichm¨uller Geodesics Mary Rees August 3, 2018 Abstract A proof of the Ending Laminations Theorem is given which uses Te- ichm¨uller geodesics directly. 1 Introduction Over 30 years ago, Scott proved the remarkable result [52], [53] that any finitely- generated fundamental group of any three-manifold N is finitely presented, and, further, that the manifold is homotopy equivalent to a compact three- dimensional submanifold with boundary. This submanifold, known as the Scott core, was shown [31] to be unique up to homeomorphism — but not, in general, up to isotopy. If, in addition, N is a hyperbolic manifold, then it is useful to consider the manifold with boundary Nd obtained by removing from N compo- nents of the ε0-thin part round cusps, for a fixed Margulis constant ε0. Sullivan [56] proved that there are only finitely many cusps. We shall call Nd the horoball deletion of N. Topologically, it does not depend on the choice of Margulis con- stant ε0, although metrically it does. There is a relative version of Scott’s result [29] which says that (N ,∂N ) is homotopy equivalent to (N ,N ∂N ) where d d c c ∩ d Nc Nd is the relative Scott core. The relative Scott core is, again, unique up to⊂ homeomorphism [30]. The closure of each component of ∂N ∂N is c \ d an orientable compact surface Sd with boundary, and is also the boundary in Nd of U(Sd), for a unique component U(Sd) of Nd Nc. The surface Sd can be any compact orientable surface with boundary apart\ from the closed disc, arXiv:math/0404007v5 [math.GT] 18 Jul 2007 annulus or torus. The set U(Sd) = U(e) is a neighbourhood of a unique end e of Nd [5]. The correspondence between end e, and the surface Sd(e) bounding its neighbourhood, is thus one-to-one. We shall also write S(e) for the surface without boundary obtained up to homeomorphism by attaching a punctured disc to each boundary component of Sd(e). In 2004, following some close approaches (e.g. [55]), two proofs of the Tame- ness conjecture were announced [2], [14], that Nc can be chosen so that each closed end neighbouhood S (e) U(e) is homeomorphic to S (e) [0, ), that d ∪ d × ∞ is, each end of Nd is topologically tame. (See also [54].) The topology of N is thus uniquely determined by the topology of the pair (N ,N ∂N ). c c ∩ d 1 The history of the geometry of N runs parallel to this. The concept of geometric tameness was originally developed by Thurston [58]. An end of Nd is geometrically finite if it has a neighbourhood disjoint from the the set of closed geodesics in N. (There are a number of equivalent definitions.) An end of Nd is simply degenerate if each neighbourhood of the end contains a simple closed geodesic. These two possibilities are mutually exclusive, and if one or the other holds, the end is said to be geometrically tame. It is not clear a priori that one or the other possibilities must hold, but in 1986 Bonahon [5] published a proof that an end e of Nd is geometrically tame if the inclusion of Sd(e) in Nc is injective on fundamental groups, that is, if Sd(e) is incompressible in Nd. Importantly for future developments, this made rigorous the end invariant 3 suggested by Thurston. In the geometrically finite case, if N = H /π1(N), and 3 Ω ∂H is the domain of discontinuity of π1(N), then the closure of U(e) in ⊂3 (H Ω/π1(N) intersects a unique punctured hyperbolic surface homeomorphic to S(∪e), and contains a subsurface which is isotopic in (H3 Ω)/π (N) to S (e). ∪ 1 d Thus, this surface in Ω/π1(N) determines a point µ(e) in the Teichm¨uller space (S(e)). Bonahon’s work dealt with the simply degenerate ends. He showed thatT if S(e) was endowed with any complete hyperbolic structure, then the Hausdorff limit of any sequence of simple closed geodesics exiting e was the same arational geodesic lamination, independent, up to homeomorphism, of the hyperbolic metric chosen. A geodesic lamination on S is a closed set of nonintersecting geodesics on S, and a lamination is arational if every simple closed loop intersects a recurrent leaf in the lamination transversally. So in both cases, geometrically finite and infinite, an invariant µ(e) is obtained, and if there are n ends ei,1 i n, this gives an invariant (µ(e1), µ(en)). Bonahon also showed that geometric≤ ≤ tameness implies topological tameness.··· Later, Canary [16] extended Bonahon’s result about geometric tameness to any end of any topologically tame N. The resolutions of the Tameness Conjecture mentioned above means that any end of any three-dimensional hyperbolic manifold with finitely generated fundamental group is topologically tame, and hence, by the work of Bonahon and Canary, geometrically tame. Bonahon’s result on an invariant (µ(e1, µ(ei)) associated to N for which ∂N ∂N is incompressible, generalises to any··· tame hyperbolic manifold N. Let c \ d G denote the group of orientation-preserving homeomorphisms ϕ of Nc, modulo isotopy, and are homotopic in N to the identity. Then G acts on (S(e)) c e T ∪ (S(e)), where the product is over ends e of Nd and (S(e)) isQ the space of geodesicGL laminations on S(e). We write [µ , µ ] to denoteGL an element of the 1 ··· n quotient space. Note that G is trivial if each Sd(e) is incompressible in Nd. We also write a(S) for the space of arational laminations in (S). The ending invariant ofGL a tame hyperbolic manifold is now a point GL n [µ(e ), µ(e )] ( (S(e )) (S(e ),N))/G 1 ··· n ∈ T i ∪ Oa i Yi=1 Here, (S,N) (S) is the Masur domain as defined by Otal [45], since the conceptO arose in⊂ the GL case of handlebodies in work by H. Masur [26]. The precise 2 definition will be given later, but (S,N) is open in (S), invariant under the action of the group G (as above),O contains no closedGL geodesics in S which are trivial in N, and (S,N)= ( ) if (S ,∂S ) is incompressible in (N ,∂N ). O GL S d d d d As for (S), a(S,N) denotes the set of arational laminations in (S,N). TheGL obviousO questions to ask about the invariants [µ(e ), µ(e O)] of hyper- 1 ··· n bolic manifolds of a fixed topological type are: does the invariant [µ(e1, µ(en)] of a hyperbolic manifold uniquely determine that manifold up to isometr··· y, and which invariants can occur. There is a natural compactification of (S), or, more generally, of (S(e))/G, in which (S(e))/G isT the interior and (S(e),N)/G e T e T e Oa Qis contained in the boundary.Q The topology at the boundaryQ will be specified later. The boundary is actually larger than (S(e),N)/G. It also includes e Oa countably many split boundary pieces, one forQ each multicurve Γ in the Masur domain. (Multicurves are defined in 2.1, and the Masur domain in 3.10.) Given a multicurve Γi on S(ei) for each end ei of N, we define a topological manifold with boundary N (Γ , Γ ) which contains N and is contained in N, as follows. d 1 ··· n c We can assume that γ Sd(ei) for all γ Γi. Fix a homeomorphism Φi from S(e ) [0, ) to the closed⊂ neighbourhood∈ of e in N bounded by S(e ). Choose i × ∞ i i a collection of disjoint open annulus neighbourhoods A(γ) Sd(ei) of the loops γ Γ . Then ⊂ ∈ i N (Γ , Γ )= N n Φ ( A(γ) (0, )), d 1 ··· n d \ ∪i=1 i ∪γ∈Γi × ∞ N (Γ , Γ )= N N (Γ , Γ ). c 1 ··· n c ∩ d 1 ··· n The closures of components of ∂Nc(Γ1, Γn) ∂Nd(Γ1, Γn) are sets Sd(ei, α), with interior homeomorphic to α, for··· each component\ ···α of S(e ) ( Γ ). We i \ ∪ i write Σ(Γi) for the set of such components. We let S(ei, α) be the topological surface obtained by adding a punctured disc round each boundary component. Write (S, Γ) = ( (S(α)) (S(α),N). W T ∪ Oa α∈YΣ(Γ The boundary of (S(e))/G consists of e T Q ( (S(e),N) (S(e), Γ))/G, Oa ∪ ∪ΓW Ye where the union is over all nonempty multicurves Γ as above. We continue to denote an element of this space by µ, so that points in (S)/G and its boundary are denoted by µ. T The Ending Laminations Theorem can then be formulated as follows, given the proofs of the Tameness conjecture. It says that the ending lamination in- variants are unique. This is the first main result of this paper. Theorem 1.1. Let N be any three-dimensional hyperbolic manifold with finitely generated fundamental group such that Nd has ends ei, 1 i n. Then N is uniquely determined up to isometry by its topological typ≤e and≤ the ending lamination data [µ(e ), ,µ(e )]. 1 ··· n 3 As for existence, we have the following, which has extensive overlap with earlier results in the literature. Theorem 1.2. Let N be any geometrically finite three-dimensional hyperbolic manifold with finitely generated fundamental group such that the horoball dele- ′ ′ tion Nd has ends ei, 1 i n.
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