Strong convergence of Kleinian groups: the cracked eggshell

TL;DR

This paper provides a comprehensive description of the space of discrete faithful representations of hyperbolizable 3-manifolds, introducing new coordinates to analyze its topology and group actions.

Contribution

It introduces extended coordinates on SH(M) and characterizes the set of representations in terms of end invariants, advancing understanding of their topological structure.

Findings

SH(M) is locally connected.

Introduces extended Ahlfors-Bers coordinates.

Analyzes the modular group's action on SH(M).

Abstract

In this paper we give a complete description of the set of discrete faithful representations SH(M) uniformizing a compact, orientable, hyperbolizable 3-manifold M with incompressible boundary, equipped with the strong topology, with the description given in term of the end invariants of the quotient manifolds. As part of this description, we introduce coordinates on SH(M) that extend the usual Ahlfors-Bers coordinates. We use these coordinates to show the local connectivity of SH(M) and study the action of the modular group of M on SH(M).