Strong convergence of Kleinian groups: the cracked eggshell

Abstract

In this paper we give a complete description of the set $\mathrm{SH}({\pi }_1(M))$ of discrete faithful representations of the fundamental group of a compact, orientable, hyperbolizable 3-manifold 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 $\mathrm{SH}({\pi }_1(M))$ that extend the usual Ahlfors–Bers coordinates. We use these coordinates to show the local connectivity of $\mathrm{SH}({\pi }_1(M))$ and study the action of the modular group of $M$ on $\mathrm{SH}({\pi }_1(M))$.