Geometry, Rigidity, and Group Actions
Geometry, Rigidity, and Group Actions
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DOI：
10.7208/chicago/9780226237909.001.0001
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
B. Farb;D. Fisher;R. Zimmer
中科院分区：
文献类型：
--
作者：
B. Farb;D. Fisher;R. Zimmer
群体行为的研究已经有一百多年的历史了，但直到今天，它仍然是一个充满活力的、在各种数学领域被广泛研究的话题。在过去的五十年里，一个中心的发展是刚性现象，由此人们可以对某些群的行为进行分类，例如半单李群中的格。这提供了一种方法来分类重要空间的所有可能的对称性以及所有允许给定对称性的空间。典型的结果可以在乔治·莫斯托、格雷戈里·马古利斯和罗伯特·J·齐默等人的开创性工作中找到。《几何、刚性和群作用》一文探讨了群作用和刚性在几个数学领域中的作用，包括遍历理论、动力学、几何、拓扑学和表示簇的代数性质。在某些情况下，可能的集体行动的动态是调查的主要焦点。在其他情况下，群作用的动力学是证明关于代数、几何或拓扑的定理的工具。这一卷包含了该领域的一些主要方向的调查，以及关于当前感兴趣的主题的研究文章。
The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gregory Margulis, and Robert J. Zimmer, among others. The papers in "Geometry, Rigidity, and Group Actions" explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field as well as research articles on topics of current interest.