Descriptive Set Theory and Measured Group Theory

描述集合论和测度群论

• 批准号: 1600904
• 负责人: Robin Tucker-Drob
• 金额: $ 15万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600904&HistoricalAwards=false
• 关键词: Descriptive; Set; Theory; Measured; Group

This project aims to address several problems in the areas of descriptive combinatorics and orbit equivalence. These topics lie at the interface of descriptive set theory, measured group theory, graph theory, ergodic theory, probability theory, and operator algebras. In recent years, mathematicians in these fields have come to realize that problems concerning algebraic, dynamical, and descriptive structural complexity of countable group and equivalence relations can be fruitfully studied via descriptive combinatorial and graph theoretic means. The principal investigator and collaborators have employed this combinatorial perspective to create new tools used to answer several open problems in these fields. In addition to a combinatorial perspective, research in these fields is also facilitated by a global perspective, from which problems in ergodic theory, for example, may be seen as topological-dynamical and descriptive problems concerning actions of the group of automorphisms of a standard probability space. This research project aims to generate more new general tools and to promote fruitful interactions among these fields. Measured group theory seeks to understand countably infinite groups through ergodic theoretic properties of their measurable actions on standard probability spaces, and particularly through structural properties of the orbit equivalence relations generated by these actions. The motivating phenomenon is that algebraic properties of an acting group are often expressed through measurable properties of the associated equivalence relation. An extreme form of this phenomenon is seen in orbit equivalence and cocycle rigidity and superrigidity theorems, which state that in certain settings an equivalence relation completely remembers the group from which it was generated. At the other extreme are what might be called orbit equivalence anti-rigidity theorems, stating that certain groups and actions cannot be distinguished by looking at the equivalence relations they generate. This research project touches upon phenomena at both of these extremes. The questions are motivated by rigidity/anti-rigidity results established by the principal investigator and collaborators, which have led to new questions addressed in this project. The project will address problems concerning algebraic, dynamical, and descriptive structural complexity of countable groups and equivalence relations, specifically in the areas of orbit equivalence, treeability, cocycle superrigidity, weak equivalence rigidity, and measurable combinatorial properties and parameters of graphs and group actions. The project pursues a descriptive combinatorial and graph theoretic approach to these problems.

本计画旨在解决描述性组合学与轨道等价领域的几个问题。这些主题在于接口的描述集理论，测量群理论，图论，遍历理论，概率论和算子代数。近年来，数学家们逐渐认识到，可数群的代数复杂性、动力复杂性和描述结构复杂性以及等价关系等问题可以通过描述组合和图论的方法得到有效的研究。主要研究者和合作者采用这种组合的观点来创建新的工具，用于回答这些领域中的几个开放问题。除了组合的角度，这些领域的研究也是由一个全球性的角度，从遍历理论的问题，例如，可以被看作是拓扑动力学和描述性的问题有关的行动组的自同构的一个标准的概率空间。该研究项目旨在产生更多新的通用工具，并促进这些领域之间富有成效的互动。测度群理论试图通过可数无限群在标准概率空间上的可测作用的遍历理论性质，特别是通过这些作用产生的轨道等价关系的结构性质来理解可数无限群。激励的现象是，作用群的代数性质通常通过相关等价关系的可测量性质来表达。这种现象的一个极端形式见于轨道等价和上循环刚性与超刚性定理，它们指出在某些情况下，等价关系完全记住了它所产生的群。另一个极端是所谓的轨道等价反刚性定理，它指出某些群和作用不能通过观察它们产生的等价关系来区分。这个研究项目涉及这两个极端的现象。这些问题的动机是由主要研究者和合作者建立的刚性/反刚性结果，这导致了本项目中解决的新问题。该项目将解决有关可数群和等价关系的代数，动力学和描述性结构复杂性的问题，特别是在轨道等价，Trestrom，cocycle superrigidity，弱等价刚性以及可测量的组合性质和参数的图和群作用等领域。该项目追求一个描述性的组合和图论方法来解决这些问题。