Institute for Mathematical Sciences

Preprint ims91-23

L. Keen, B. Maskit, and C. Series
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 $\Omega(G)$ 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.

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