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1. the Ending Lamination Conjecture 1 2 THE CLASSIFICATION OF KLEINIAN SURFACE GROUPS, II: THE ENDING LAMINATION CONJECTURE JEFFREY F. BROCK, RICHARD D. CANARY, AND YAIR N. MINSKY Abstract. Thurston's Ending Lamination Conjecture states that a hy- perbolic 3-manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups. The main ingredi- ent is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in [47], and a sub- sequent paper [15] will establish the Ending Lamination Conjecture in general. Contents 1. The ending lamination conjecture 1 2. Background and statements 8 3. Knotting and partial order of subsurfaces 25 4. Cut systems and partial orders 49 5. Regions and addresses 66 6. Uniform embeddings of Lipschitz surfaces 76 7. Insulating regions 91 8. Proof of the bilipschitz model theorem 99 9. Proofs of the main theorems 118 10. Corollaries 119 References 123 1. The ending lamination conjecture In the late 1970's Thurston formulated a conjectural classification scheme for all hyperbolic 3-manifolds with finitely generated fundamental group. The picture proposed by Thurston generalized what had hitherto been un- derstood, through the work of Ahlfors [3], Bers [10], Kra [34], Marden [37], Maskit [38], Mostow [51], Prasad [55], Thurston [67] and others, about geo- metrically finite hyperbolic 3-manifolds. Thurston's scheme proposes end invariants which encode the asymptotic geometry of the ends of the manifold, and which generalize the Riemann Date: December 7, 2004. Partially supported by NSF grants DMS-0354288, DMS-0203698 and DMS-0203976. 1 2 JEFFREY F. BROCK, RICHARD D. CANARY, AND YAIR N. MINSKY surfaces at infinity that arise in the geometrically finite case. Thurston made this conjecture in [67]: Ending Lamination Conjecture. A hyperbolic 3-manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. This paper is the second in a series of three which will establish the Ending Lamination Conjecture for all topologically tame hyperbolic 3-manifolds. (For expository material on this conjecture, and on the proofs in this paper and in [47], we direct the reader to [43],[48] and [49]). Together with the recent proofs of Marden's Tameness Conjecture by Agol [2] and Calegari-Gabai [17], this gives a complete classification of all hyperbolic 3-manifolds with finitely-generated fundamental group. In this paper we will discuss the surface group case. A Kleinian sur- face group is a discrete, faithful representation ρ : π1(S) PSL2(C) where S is a compact orientable surface, such that the restriction! of ρ to any boundary loop has parabolic image. These groups arise naturally as re- strictions of more general Kleinian groups to surface subgroups. Bonahon [11] and Thurston [66] showed that the associated hyperbolic 3-manifold 3 Nρ = H /ρ(π1(S)) is homeomorphic to int(S) R and that ρ has a well- × defined pair of end invariants (ν+; ν ). Typically, each end invariant is either a point in the Teichm¨ullerspace of−S or a geodesic lamination on S. In the general situation, each end invariant is a geodesic lamination on some (possi- bly empty) subsurface of S and a conformal structure on the complementary surface. We will prove: Ending Lamination Theorem for Surface Groups A Kleinian surface group ρ is uniquely determined, up to conjugacy in PSL2(C), by its end invariants. The first part of the proof of this theorem appeared in [47], and we will refer to that paper for some of the background and notation, although we will strive to make this paper readable fairly independently. See Section 1.3 for a discussion of the proof of the general Ending Lami- nation Conjecture, which will appear in [15]. Bilipschitz Model Theorem. The main technical result which leads to the Ending Lamination Theorem is the Bilipschitz Model Theorem, which gives a bilipschitz homeomorphism from a \model manifold" Mν to the hyperbolic manifold Nρ (See 2.7 for a precise statement). The model Mν was constructed in Minsky [47],x and its crucial property is that it depends only on the end invariants ν = (ν+; ν ), and not on ρ itself. (Actually Mν − is mapped to the \augmented convex core" of Nρ, but as this is the same as Nρ in the main case of interest, we will ignore the distinction for the rest of the introduction. See 2.7 for details.) x THE CLASSIFICATION OF KLEINIAN SURFACE GROUPS, II 3 The proof of the Bilipschitz Model Theorem will be completed in Section 8, and the Ending Lamination Conjecture will be obtained as a consequence of this and Sullivan's rigidity theorem in Section 9. In that section we will also prove the Length Bound Theorem, which givesn estimates on the lengths of short geodesics in Nρ (see 2.7 for the statement). x 1.1. Corollaries A positive answer to the Ending Lamination Conjecture allows one to settle a number of fundamental questions about the structure of Kleinian groups and their deformation spaces. In the sequel, we will see that it gives (together with convergence theorems of Thurston, Ohshika, Kleineidam- Souto and Lecuire) a complete proof of the Bers-Sullivan-Thurston density conjecture, which predicts that every finitely generated Kleinian group is an algebraic limit of geometrically finite groups. In the surface group case, the density conjecture follows immediately from our main theorem and results of Thurston [65] and Ohshika [52]. We recall that AH(S) is the space of conjugacy classes of Kleinian surface groups and that a surface group is quasifuchsian if both its ends are geometrically finite and it has no additional parabolic elements. Density Theorem. The set of quasifuchsian surface groups is dense in AH(S). Marden [37] and Sullivan [63] showed that the interior of AH(S) consists exactly of the quasifuchsian groups. Bromberg [16] and Brock-Bromberg [14] previously showed that freely indecomposable Kleinian groups without parabolic elements are algebraic limits of geometrically finite groups, using cone-manifold techniques and the bounded-geometry version of the Ending Lamination Conjecture in Minsky [46]. We also obtain a quasiconformal rigidity theorem that gives the best possible common generalization of Mostow [51] and Sullivan's [62] rigidity theorems. Rigidity Theorem. If two Kleinian surface groups are conjugate by an orientation-preserving homeomorphism of C, then they are quasiconformally conjugate. b We also establish McMullen's conjecture that the volume of the thick part of the convex core of a hyperbolic 3-manifold grows polynomially. thick If x lies in the thick part of the convex core CN , then let BR (x) be the set of points in the 1-thick part of CN which can be joined to x by a path of length at most R lying entirely in the 1-thick part. Volume Growth Theorem. If ρ : π1(S) PSL2(C) is a Kleinian surface 3 ! group and N = H /ρ(π1(S)), then for any x in the 1-thick part of CN , volume Bthick(x) cRd(S); R ≤ 4 JEFFREY F. BROCK, RICHARD D. CANARY, AND YAIR N. MINSKY where c depends only on the topological type of S, and χ(S) genus(S) > 0 d(S) = (− χ(S) 1 genus(S) = 0: − − A different proof of the Volume Growth Theorem is given by Bowditch in [13]. We are grateful to Bowditch for pointing out an error in our original definition of d(S). Proofs of these corollaries are given in Section 10. Each of them ad- mit generalizations to the setting of all finitely generated Kleinian groups and these generalizations will be discussed in [15]. In that paper, we will also use the solution of the Ending Lamination Conjecture and work of Anderson-Canary-McCullough [7] to obtain a complete enumeration of the components of the deformation space of a freely indecomposable, finitely generated Kleinian group. Another corollary, which we postpone to a separate article, is a theorem which describes the topology and geometry of geometric limits of sequences of Kleinian surface groups { in particular such a limit is always homeomor- phic to a subset of S R. This was also shown by Soma [58]. × 1.2. Outline of the proof The Lipschitz Model Theorem, from [47], provides a degree 1 homotopy equivalence from the model manifold Mν to the hyperbolic manifold Nρ (or in general to the augmented core of Nρ, but we ignore the distinction in this outline), which respects the thick-thin decompositions of Mν and Nρ and is Lipschitz on the thick part of Mν. (See 2.7.) Our main task in this paper is to promotex this map to a bilipschitz home- omorphism between Mν and Nρ , and this is the content of our main result, the Bilipschitz Model Theorem. The proof of the Bilipschitz Model Theorem converts the Lipschitz model map to a bilipschitz map incrementally on var- ious subsets of the model. The main ideas of the proof can be summarized as follows: Topological order of subsurfaces. In section 3 we discuss a \topological order relation" among embedded surfaces in a product 3-manifold S R. This is the intuitive notion that one surface may lie \below" another in× this product, but this relation does not in fact induce a partial order and hence a number of sticky technical issues arise. We introduce an object called a “scaffold”, which is a subset of S R consisting of a union of unknotted solid tori and surfaces in S R, each× isotopic to a level subsurface, satisfying certain conditions. The× main the- orem in this section is the Scaffold Extension Theorem (3.8), which states that, under appropriate conditions (in particular an \order-preservation" condition), embeddings of a scaffold into S R can be extended to global homeomorphisms of S R.
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