FRG: Asymptotic and probabilistic methods in geometric group theory

该项目的主题是适宜性及其相关概念(Kazhdan Property T、Property tau、Unitarizable等)。以及它在从数论到拓扑学和泛函分析的不同数学领域中的应用。特别是，PI将集中在以下问题上：*服从群和结合代数的分类，*Golod-Shafarevich群的可选性和三维拓扑和数论中的相关问题，*扩展图，SL(2，C)中格子的性质tau，群论中的概率方法，*可分解性问题，*构造具有性质T的有限表示群的新例子，*离散群和准p-群的线性，*离散群体的渐近性质。私人投资促进机构将就项目的不同方面组织几个会议和工作坊。NSF的拨款将支持在他们的指导下工作的几名毕业生和博士后研究员。群论诞生于对称论。Gauss、Abel、Galois、Lie等人的工作表明，对称群承载着有关代数方程和微分方程的可解性的基本信息。群论在数学和物理的许多领域发挥着至关重要的作用。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，群体理论从它与其他领域的联系中受益匪浅。大约80年前，von Neumann，Banach和Tarski引入了amenabile群的概念，并将其与一些基本问题联系在一起，例如：一个人能否给我们空间中的任何一组点赋权，使其在空间的所有对称下是不变的？PI将探索群和代数的适应性的各个方面，以及适应性群、数论和拓扑学之间的深层次联系。

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.