CAREER: Amenable and recurrent actions of finitely generated groups

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Abstract (Juschenko, 1352173): The subject of "amenability" essentially begins in the 1900s with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. The class of amenable groups was introduced and studied by von Neumann in 1929, and he explained why a paradox appeared only in dimensions greater or equal to three. In the 1940s the amenability theory shifted into the field of functional analysis. Currently amenability theory appears in many fields of mathematics, most notably in operator algebras, functional analysis, ergodic theory, probability theory, and harmonic analysis.The core part of the project is based on the recent technique used by PI to prove that certain groups are amenable. PI plans to develop this technique in a more conceptual way using random walks on Schreier graphs. There are several classes of groups to which the technique can potentially be applied. They include Thompson group F, interval exchange transformation group, topological full group of abelian groups. The expected impact of the project is to give deeper understanding of analytic properties of the group actions and relate them to other fields. One of the central impact of this proposal is through its educational goals, in particular, in organizing of several summer schools and workshops. The schools are aimed toward graduate students and other young researchers, focused on the recent developments in group theory and operator algebras.