Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets.
Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets.
复制标题
具有圆堆积极限集的克莱因群的几何有限性和唯一性。
DOI：
10.1515/crll.1993.436.209
复制
发表时间：
1991
期刊：
影响因子：
--
通讯作者：
L. Keen
中科院分区：
文献类型：
--
作者：
B. Maskit;L. Keen
本文假设G是一个非零生成的挠自由非初等Kleinian群，且Ω（G）非空.我们表明，最大数量的元素的$G$，可以捏正是最大数量的秩1抛物子群，任何组同构的$G$可能包含。一个具有最多秩1极大抛物子群的群称为{\it maximally parabolic}。我们发现这样的群体存在。我们在这里简明地陈述我们的主要定理。
定理I极大抛物群的极限集是一个圆填充;也就是说，它的正则集的每个分支都是一个圆盘。
定理II.一个极大抛物群是几何有限的。
定理III.一个极大抛物Pinch函数群在$PSL（2，{\bfc}）$中由它的抽象同构类和它的抛物元决定直到共轭.
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.