On topologically tame Kleinian groups with bounded geometry

Summary

Abstract

We consider a generalisation of Minsky's rigidity theorem for freely indecomposable Kleinian groups with bounded geometry to topologically tame freely decomposable case in order to apply it to the following two results. The first is the uniqueness property for the problem of realising given end invariants by a group lying on the boundary of the quasi-conformal deformation space of a convex cocompact group. The second is a generalisation of Soma's result on the third bounded cohomology groups of closed surface groups to the case of free groups.

Introduction

A topologically tame hyperbolic 3-manifold has three pieces of information: the homeomorphism type, the conformal structures at infinity for geometrically finite ends (of non-cuspidal part), and the ending laminations for geometrically infinite ends. The ending lamination conjecture, due to Thurston, says that these pieces of information uniquely determine the isometry type of the manifold. Recently, Minsky proved this conjecture affirmatively, partially collaborating with Brock and Canary. Although the result is in the process of publication, the special case when manifolds have freely indecomposable fundamental group and bounded geometry, i.e., when the injectivity radii are bounded below by a positive constant, has been already published in [Min94], [Min00] and [Min01]. In the present paper, we shall explain how the argument of Minsky there can be generalised to the case of topologically tame manifolds with bounded geometry possibly with freely decomposable fundamental groups, and then show the following two kinds of its applications.