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

Abstract

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 $\mathbb{H}^3$ by a Kleinian surface group.