Topics in Measurable Group Theory

项目编号：dms -0905977项目负责人：Alexander furman项目主要包括三个研究方向，涉及无限群体和群体行为动力学。这三个方向包括：(1)对一些高阶超刚性现象的统一方法。这种方法基于广义Weyl群的概念，应该涵盖许多已知的结果，包括原始的马古利斯超刚性及其由齐默引起的共循环版本，抽象产物的结果，作用于外来仿射建筑的群的新结果。(2)可数群间嵌套的测度论概念。该项目是研究可测双嵌入群的等价类，以及这些类之间的顺序。这些技术包括第(1)部分中得到的环超刚性结果。(3)考虑有界畸变的gromov -双曲群上不变度量的“模空间”。这种度量的指导和激励源是从固定闭流形上的负曲率黎曼度量提升而来的度量。该项目属于代数、几何和动力学相互作用的研究领域。这些项目的共同主题是研究动力学、代数和几何中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，并对这些领域最近的一些重要成果进行了可能的概括。

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.