Cores of hyperbolic 3-manifolds and limits of Kleinian groups II

TL;DR

This paper proves Troels Jorgensen's conjecture that algebraic and geometric limits of converging Kleinian groups coincide under certain conditions, especially when the algebraic limit's domain of discontinuity is non-empty.

Contribution

It establishes the conjecture in most cases, including when the algebraic limit has a non-empty domain of discontinuity, and verifies it for a broad class of finitely generated Kleinian groups.

Findings

Algebraic and geometric limits agree when the algebraic limit's domain of discontinuity is non-empty.

Limit sets of the groups in the sequence converge to the limit set of the algebraic limit.

The conjecture is verified for many finitely generated Kleinian groups, excluding certain free products.

Abstract

Troels Jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in 'most' cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty. We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups.