FRG: Asymptotic and probabilistic methods in geometric group theory

该项目的主题是可适应性和相关概念（Kazhdan属性T，属性tau，统一性等）。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将集中于以下问题： * 顺从群和结合代数的分类， * Golod-Shafarevich群的顺从性及三维拓扑和数论中的相关问题， * 扩展图，SL（2，C）中格的性质tau，群论中的概率方法， * Dixonitarizability问题， * 构造了具有性质T的群的新例子， * 离散和亲p-基团的线性， * 离散群的渐近性质。PI将组织几次会议和研讨会，讨论项目的不同方面。美国国家科学基金会的拨款将支持一些研究生和博士后研究员在他们的监督下工作。 群论作为对称性理论而诞生。高斯、阿贝尔、伽罗瓦、李和其他人的工作表明，对称群承载着关于代数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中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，群论从它与其他领域的联系中受益匪浅。大约80年前，冯·诺依曼，巴拿赫和塔斯基引入了amenabile群的概念，并将其与基本问题联系起来，如“可以给我们空间中的任何点集分配一个权重，使得权重在空间的所有对称性下不变吗？".该PI将探索群体和代数的顺从性的各个方面，以及顺从群体，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 anddifferential 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.