FRG: Asymptotic and probabilistic methods in geometric group theory

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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.