可数离散群顺从性的等价条件

A Review of Equivalent Conditions for Amenability of Countable Discrete Groups

摘要: 顺从性在动力系统中一直是一个热点课题。这一概念最初是由冯诺依曼针对离散群而提出来的，时至今日，已经拓展到数学的许多领域。但是，关于群满足顺从性的一系列等价条件在国内一直没有系统性的整理，国外相关文献又存在定理证明简略难懂、文献分散的情况。并且这些条件之间的联系国内外均未提及。因此，本文旨在系统完整的梳理群满足顺从性的相关等价条件并分析它们之间的内在联系。通过文献研究法梳理了可数离散群满足顺从性的常见等价条件，包括存在正定函数点点逼近单位函数、Følner条件和Reiter性质等。在此基础上，分析了它们之间的内在联系并得到关系图。

Abstract: Amenability has always been a hot topic in dynamical systems theory. This concept was first proposed by von Neumann for discrete groups, which has been extended to many areas of mathematics. However, the equivalent conditions of amenability have not been systematically sorted out in China, and the proof of relevant theorems in foreign literature is simple and difficult to understand and scattered. Besides, the relations between them are also not mentioned. Therefore, the purpose of this paper is to systematically and completely comb out the equivalence conditions of compliance and analyze the internal relations between them. We get the common equivalent conditions of the amenability for countable discrete groups by literature research, including the positive definite function tend pointwise to the unit function, Følner condition, Reiter property and so on. On this basis, we analyze the internal relations between them and get the relationship diagram between them.