The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

Abstract

Thurston's Ending Lamination Conjecture states that a hyperbolic 3manifold 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. Contents 1. The Ending Lamination Conjecture 1 2. Background and statements 10 3. Scaffolds and partial order of subsurfaces 30 4. Cut systems and partial orders 55 5. Regions and addresses 73 6. Uniform embeddings of Lipschitz surfaces 83 7. Insulating regions 103 8. Proof of the bilipschitz model theorem 113 9. Proof of the Ending Lamination Theorem 134 10. Corollaries 137 References 144 1. The Ending Lamination Conjecture