Convergence of freely decomposable Kleinian groups

TL;DR

This paper proves that sequences of geometrically finite hyperbolic 3-manifolds with diverging boundary structures converge under certain conditions, extending Thurston's double limit theorem and confirming a conjecture on function group convergence.

Contribution

It generalizes Thurston's double limit theorem and confirms a conjecture on the convergence of function groups for hyperbolic 3-manifolds with doubly incompressible laminations.

Findings

Sequences with doubly incompressible boundary laminations have compact closure in deformation space.

Generalization of Thurston's double limit theorem.

Affirmative proof of a conjecture on convergence of function groups.

Abstract

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressible, then the sequence has compact closure in the deformation space. As a consequence we generalise Thurston's double limit theorem and solve his conjecture on convergence of function groups affirmatively.