On Poincaré's theorem for fundamental polygons
On Poincaré's theorem for fundamental polygons
复制标题
关于基本多边形的庞加莱定理
DOI：
10.1016/s0001-8708(71)80003-8
复制
发表时间：
1971
影响因子：
1.7
通讯作者：
B. Maskit
中科院分区：
文献类型：
--
作者：
B. Maskit
庞加莱关于基本多边形的经典定理[2]给出了多边形是Fuchsian群的基本域的充分条件。这个定理有几个已发表的证明，但它们的有效性存在一些问题；Siegel[4，第115页]对此进行了评论，并在相当严格的条件下给出了一个明显有效的证明。所有已发表的证明都不像它们可能的那样普遍，它们都有一个从未真正使用过的凸性条件。本照会试图澄清情况。下面提出的问题和解决办法是在几次非正式谈话中产生的。在场的有LV Ahlfors、L. Bers、W. Magnus、JE McMillan和B. Maskit。庞卡德也发表了这个定理的推广。推广到三维双曲空间中的多面体，其中不连续群是Kleinian而不是Fuchsian。阿尔伯特·马登（Albert Marden）最近的工作表明了这些多面体对克莱因群研究的重要性，因此我附上了对庞加莱多面体定理的陈述和证明。
Poincar6's classical theorem on fundamental polygons [2] gives sufficient conditions for a polygon to be a fundamental domain for a Fuchsian group. There are several published proofs of this theorem, but there is some question as to their validity; Siegel [4, p. 115] has commented on this and given an apparently valid proof under fairly restrictive conditions. None of the published proofs are as general as they might be, and they all have a convexity condition that is never really used.This note is an attempt to clarify the situation. The problem and the solution presented below arose during the course of several informal conversations. Present at one or more of these conversations were LV Ahlfors, L. Bers, W. Magnus, JE McMillan, and B. Maskit. Poincard [3] also published a generalization of this theorem. The generalization is to polyhedra in 3-dimensional hyperbolic space, where the discontinuous group is Kleinian rather than Fuchsian. The recent work of Albert Marden [1] shows the importance of these polyhedra for the study of Kleinian groups, and so I have appended a statement and proof of Poincar6's polyhedron theorem.