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

From WebRef.org
Jump to navigationJump to search

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.

https://arxiv.org/abs/math/9201299


Sponsor: The Simba Duvet, designed using space-inspired fabric to regulate your temperature while you sleep. Shop Now.

Up to 60%