Generic fundamental polyhedra for kleinian groups
Generic fundamental polyhedra for kleinian groups
复制标题
克莱因群的通用基本多面体
DOI：
10.1007/978-1-4613-9611-6_7
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
A. Marden
中科院分区：
文献类型：
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作者：
T. Jørgensen;A. Marden
对于研究克莱因群的基本多面体的爱好者来说，能够在每个特定情况下选择一个具有最简单的边和顶点局部结构的多面体是有帮助的。例如，在研究[4]中的小变形时，将一个基团的基本多面体与附近基团的基本多面体进行比较；如果一个多面体是尽可能简单的，那么附近的多面体也会趋于简单。本文的目的就是要找到这样的多面体。事实上，我们将证明给定群的“一般”基本多面体在群的代数/几何结构允许的情况下是尽可能简单的。当群中没有椭圆变换时，可以精确地识别出一般多面体的特征。另一方面，在有扭转的群中，椭圆变换的某些构型，例如三个轴成对共面的椭圆，涉及到额外的困难，我们决定把这些情况放在一边。
For aficionados of fundamental polyhedra in the study of the kleinian groups, it is helpful to be able to choose in each particular case a polyhedron with the simplest possible local structure about its edges and vertices. For example, in the study of small deformations as in [4], a fundamental polyhedron for one group is compared to those of nearby groups; if the one polyhedron is as simple as possible, the nearby ones will tend to be as well. It is the purpose of the present note to find such polyhedra. Indeed, we will show that the “generic” fundamental polyhedra for a given group are as simple as the algebraic/geometric structure of the group allows. When the group has no elliptic transformations, the features of the generic polyhedra will be precisely identified. In groups with torsion, on the other hand, certain configurations of elliptic transformations, for example three elliptics whose axes are pairwise coplanar, involve additional difficulties and we have decided to leave these cases aside.