SOME REMARKS ON NON-DISCRETE MÖBIUS GROUPS
SOME REMARKS ON NON-DISCRETE MÖBIUS GROUPS
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关于非离散莫比乌斯群的一些评论
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
A. Beardon
中科院分区：
文献类型：
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作者：
A. Beardon
本文对SL（2，R）的非离散子群的结构作了一些尝试性的描述。其主要思想是，如果一个单参数群族Gz随参数z解析变化，那么，使用解析延拓，关于离散群的某些结果可以解析延拓到族中不离散的群。该文件集中在家庭的群体所产生的两个抛物变换，并作为一个例子，包含一个证明，但一个可数集的例外值的参数z，双曲面积的双曲quadilateral双方配对的一些对抛物发电机的Gz是独立的选择发电机。这是熟悉的结果，即离散群的基本区域的面积与生成元的选择无关，但它适用于几乎所有的非离散群。它还表明，存在例外的群体，并给出了明确的例子。论文以一些未回答的问题结束。
This paper contains some tentative steps towards describing the structure of non-discrete subgroups of SL(2,R) . The main idea is that if a one-parameter family of groups Gz varies analytically with the parameter z , then, using analytic continuation, certain results about discrete groups can be analytically continued to those groups in the family that are not discrete. The paper concentrates on families of groups generated by two parabolic transformations and, as an illustration, contains a proof that, for all but a countable set of exceptional values of the parameter z , the hyperbolic area of a hyperbolic quadilateral whose sides are paired by some pair of parabolic generators of Gz is independent of the choice of generators. This is the analogue of the familiar result that the area of a fundamental region of a discrete group is independent of the choice of the generators, but it applies here to almost all non-discrete groups in the family. It is also shown that exceptional groups exist, and explicit examples of these are given. The paper ends with some unanswered questions.