Coarse Geometry of Topological Groups
Coarse Geometry of Topological Groups
复制标题
拓扑群的粗略几何
DOI：
10.1017/9781108903547
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发表时间：
2021
影响因子：
0.7
通讯作者：
Christian Rosendal
中科院分区：
文献类型：
--
作者：
Christian Rosendal
本书提供了数学实践中出现的许多非局部紧拓扑变换群的几何群论的一般框架，包括流形的同胚和微分同构群，可分度量空间的等距群和可数结构的自同构群。利用Roe的粗糙结构和空间框架，在所有拓扑群上定义了一个自然的粗糙几何结构。这种结构可用于研究，特别是在波兰群的情况下，并且通常具有明确的描述，在熟悉的情况下推广众所周知的结构，包括有限生成离散群，紧生成局部紧群和Banach空间。在大多数情况下，粗糙的几何结构是可度量的，甚至可以细化为群上的正则拟度量结构。这本书包含了许多工作的例子和足够的介绍性材料，以方便开始研究生。附录概述了这个年轻而丰富的理论中几个有待解决的问题。
This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.