Analytic Properties of Group Actions of Finitely Generated Groups

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The core of the project is based on the recent techniques developed by PI and aimed to deepen the understanding of several analytic properties of groups and their actions. One of the central notions of the research is amenability, which relates to an averaging operation that is invariant under translation by group elements. It naturally appears in many fields of mathematics: operator algebras, functional analysis, ergodic theory, probability theory, harmonic analysis. Amenability has many applications and considerable effort has been given throughout the years to showing that various groups are amenable. The PI will study a newly developed property of group actions: Liouville property. The property can be used to prove non-amenability due to the fact that if a group admits a non-Liouville action, then the group is not amenable. The PI have already developed tools for verifying Liouville property for graphs of the actions. These tools brought a connection of Liouville property of actions for strongly transitive groups and additive combinatorics. The project describes several questions on Liouville actions with streamlined approaches, as well as more ambitious problems and conjectures. In particular, the PI outlines an approach to non-amenability of Thompson group F, which is a widely open problem in the group theory. The PI provides an approach to amenability of interval exchange transformation groups. The actions of these are Liouville, moreover, they are completely Liouville. Thus, amenability of these groups can not be approached using Liouville property. For this reason, the PI will study a property of random walk of action: growth of inverted orbits. Using this property there is a potential to determine amenability of the interval exchange transformation group. On the other hand the study of this property property leads to a more general theory that is planned to be developed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.