FRG: Asymptotic and probabilistic methods in geometric group theory

本项目的主题是适应性和相关概念（哈萨克性质T、性质tau、一元性等）及其在数论、拓扑和泛函分析等不同数学领域的应用。特别需要指出的是，ppi将集中解决以下问题：*扩展图，SL（2，C）中格的性质tau，群论中的概率方法，* Dixmier酉化问题，*构造具有性质T的有限呈现群的新例子，*离散群和亲-群的线性性，*离散群的渐近性质。项目负责人将就项目的不同方面组织几次会议和讲习班。NSF的资助将支持几名研究生和博士后在他们的指导下工作。群论是作为对称理论诞生的。高斯、阿贝尔、伽罗瓦、李等人的工作表明，对称群携带着代数方程和微分方程可解性的基本信息。群论在数学和物理的许多领域起着至关重要的作用。此外，群论的最新进展表明，数学的许多领域是密切相关的。反过来，群论也从它与其他领域的联系中获益匪浅。大约80年前，冯·诺伊曼、巴拿赫和塔斯基提出了可服从群的概念，并将其与一些基本问题联系起来，比如“能否给空间中的任何点集合赋予一个权重，使其在空间的所有对称性下都不变？”这些课程将探讨群和代数的可适性的各个方面，以及可适群、数论和拓扑学之间的深层联系。

The main theme of this project is amenability and related concepts(Kazhdan property T, property tau, unitarizability, etc.) and itsapplications in different areas of mathematics from number theory totopology and functional analysis. In particular, the PIs willconcentrate on the following problems: * Classification of amenable groups and associative algebras, * Amenability of Golod-Shafarevich groups and related problems in3-dimensional topology and number theory, * Expander graphs, property tau for lattices in SL(2,C),probabilistic methods in group theory, * The Dixmier Unitarizability problem, * Constructing new examples of finitely presented groups withproperty T, * Linearity of discrete and pro-p-groups, * Asymptotic properties of discrete groups.The PIs are going to organize several conferences and workshops on differentaspects of the projects. The NSF grant will support several graduatestudents and postdoctoral fellows working undertheir supervision. Group theory was born as the theory of symmetry. The work of Gauss,Abel, Galois, Lie and others showed that groups of symmetries carry essential information about solvability of algebraic and differential equations. Group theory plays crucial role in many areas of mathematics and physics. Moreover, recent advances in group theory showed that many areas of mathematics are closely related. In turn, group theory has benefited tremendously from its connections with other areas. About 80 years ago, von Neumann, Banach and Tarski introduced the concept of amenabile group and connected it with basic questions like "Can one assign a weight to any set of points in our space so that the weight is invariant under all symmetries of the space?". The PIs will explore various aspects of amenability of groups and algebras, and deep connections between amenabile groups, number theory and topology.