Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N 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 general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is a uniformly bilipschitz model for the quotient of H 3 by a Kleinian surface group. The first half of the proof appeared in [54], and a subsequent paper [18] will establish the Ending Lamination Conjecture in general. Contents 1. The ending lamination conjecture 1 2. Background and statements 10 3. Scaffolds and partial order of subsurfaces 29 4. Cut systems and partial orders 54 5. Regions and addresses 71 6. Uniform embeddings of Lipschitz surfaces 81 7. Insulating regions 100 8. Proof of the bilipschitz model theorem 110 9. Proof of the ending lamination theorem 130 10. Corollaries 133 References 140

Abstract. Using the thermodynamic formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out(Γ)-invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group Γ into SLm(R). In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil-Petersson metric on the Fuchsian loci. Moreover, we produce Out(Γ)-invariant metrics on deformation spaces of convex cocompact representations into PSL 2 (C) and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.

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Let N = H 3 / Γ N = {{\mathbf {H}}^3}/\Gamma be a hyperbolic 3 3 -manifold which is homeomorphic to the interior of a compact 3 3 -manifold. We prove that N N is geometrically tame. As a consequence, we prove that Γ \Gamma ’s limit set L Γ {L_\Gamma } is either the entire sphere at infinity or has measure zero. We also prove that N N ’s geodesic flow is ergodic if and only if L Γ {L_\Gamma } is the entire sphere at infinity.

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