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

TL;DR

This paper proves Thurston's Ending Lamination Conjecture for Kleinian surface groups, showing that hyperbolic 3-manifolds are uniquely determined by their topological type and end invariants, using a bilipschitz model approach.

Contribution

It establishes the conjecture specifically for Kleinian surface groups, advancing the understanding of hyperbolic 3-manifolds and their classification.

Findings

Proves the Ending Lamination Conjecture for Kleinian surface groups.

Develops a uniformly bilipschitz model for these groups.

Simplifies the proof for manifolds with incompressible ends relative to cusps.

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 the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in math.GT/0302208, and a subsequent paper will establish the Ending Lamination Conjecture in general.