Tameness of hyperbolic 3-manifolds

TL;DR

This paper proves that hyperbolic 3-manifolds with finitely generated fundamental groups are tame, confirming a long-standing conjecture and impacting the understanding of Kleinian groups and 3-manifold topology.

Contribution

It establishes the tameness of hyperbolic 3-manifolds with finitely generated fundamental groups, extending results to pinched negatively curved manifolds with cusps.

Findings

Confirmed the tameness conjecture for hyperbolic 3-manifolds.

Proved the ends of these manifolds are topologically products.

Implications for the Ahlfors measure conjecture.

Abstract

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved manifolds with hyperbolic cusps. This answers a conjecture of Marden and implies the Ahlfors measure conjecture. Applications are given to other questions about Kleinian groups and 3-manifolds.