Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets

Abstract

In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $Ω(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1 parabolic subgroups that any group isomorphic to $G$ may contain. A group with this largest number of rank 1 maximal parabolic subgroups is called {\it maximally parabolic}. We show such groups exist. We state our main theorems concisely here. Theorem I. The limit set of a maximally parabolic group is a circle packing; that is, every component of its regular set is a round disc. Theorem II. A maximally parabolic group is geometrically finite. Theorem III. A maximally parabolic pinched function group is determined up to conjugacy in $PSL(2,{\bf C})$ by its abstract isomorphism class and its parabolic elements.