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

Linda Keen, Bernard Maskit, Caroline Series

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,C) by its abstract isomorphism class and its parabolic elements.