Topics in Measurable Group Theory

可测群论主题

• 批准号: 0905977
• 负责人: Alexander Furman
• 金额: $ 46.02万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905977&HistoricalAwards=false
• 关键词: Topics; Measurable; Group; Theory

AbstractAward: DMS-0905977Principal Investigator: Alexander FurmanThe proposed project contains three research directions involvinginfinite groups and dynamics of group actions. These threedirections include: (1) A uniform approach to a number of higherrank superrigidity phenomena. This approach, based on the notionof a generalized Weyl groups, should cover many known results,including the original Margulis' superrigidity, and its cocycleversion due to Zimmer, results for abstract products, new resultsfor groups acting on exotic affine buildings. (2)Measure-theoretic notion of imbedding between countable groups.The project is to study the resulting equivalence classes ofMeasurably Bi-Imbeddable groups, and the order between theseclasses. The techniques include cocycle superrigidity resultsobtained in part (1). (3) "Moduli space" of invariant metrics ona Gromov-hyperbolic group taken up to bounded distortion. Theguiding and motivating source for such metrics are metrics liftedfrom Riemannian metrics of negative curvature on a fixed closedmanifold.The project belongs to an area of research where Algebra,Geometry and Dynamics interact. The common theme of the projectis the study of intrinsic symmetries of certain structures whichappear in Dynamics, Algebra and Geometry, and the impact thatsuch intrinsic symmetries have on the studied objects. Theproposed problems suggest new points of view and possiblegeneralizations to some important recent results in these areas.

AbstractAward：DMS-0905977首席研究员：亚历山大弗曼拟议的项目包含三个研究方向涉及无限群体和群体行动的动力学。 这三个方向包括：（1）对一些高阶超刚性现象的统一处理。这种方法基于广义Weyl群的概念，应该涵盖许多已知的结果，包括原来的Margulis' superrigidity，和它的余循环版本由于Zimmer，结果抽象产品，新的resultsfor集团作用于异国情调的仿射buildings。 （2）可数群间嵌入的测度论概念，研究了可测双嵌入群的等价类，以及这些等价类之间的阶。这些技巧包括第（1）部分中得到的上循环超刚性结果。 (3)Gromov-双曲群上的不变度量的模空间。 这些度量的指导和激励来源是从固定闭流形上的负曲率黎曼度量中提升出来的度量。该项目属于代数、几何和动力学相互作用的研究领域。 该项目的共同主题是研究出现在动力学，代数和几何中的某些结构的内在对称性，以及这种内在对称性对研究对象的影响。 提出的问题提出了新的观点和可能的推广，在这些领域的一些重要的最近的结果。