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

We present a coarse interpretation of the Weil-Petersson distance d W P ( X , Y ) d_{\mathrm {WP}}(X,Y) between two finite area hyperbolic Riemann surfaces X X and Y Y using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q ( X , Y ) Q(X,Y) with X X and Y Y in its conformal boundary is comparable to the Weil-Petersson distance d W P ( X , Y ) d_{\mathrm {WP}}(X,Y) . In applications, we relate the Weil-Petersson distance to the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian for Q ( X , Y ) Q(X,Y) , and give a new finiteness criterion for geometric limits.

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends.

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