Analytic Properties of Group Actions of Finitely Generated Groups

该项目的核心是基于PI开发的最新技术，旨在加深对群体及其行动的几个分析性质的理解。研究的中心概念之一是顺从性，它涉及到一个平均操作，是不变的平移下的组元素。它自然地出现在许多数学领域：算子代数，泛函分析，遍历理论，概率论，调和分析。顺从性有许多应用，多年来，人们付出了相当大的努力来表明各种群体都是顺从的。 PI将研究群作用的一个新的性质：Liouville性质。 这个性质可以用来证明不可顺从性，因为如果一个群允许一个非刘维尔作用，那么这个群是不可顺从的。PI已经开发了用于验证动作图的Liouville性质的工具。这些工具将强传递群作用的刘维尔性质与加法组合数学联系起来。该项目以简化的方法描述了关于刘维行动的几个问题，以及更雄心勃勃的问题和建议。特别是，PI概述了汤普森群F的非顺从性的方法，这是群论中一个广泛开放的问题。PI为区间交换变换群的顺从性提供了一种方法。 他们的行为是刘维尔，而且，他们完全是刘维尔。因此，这些群体的顺从性不能用刘维尔性质来处理。因此，PI将研究作用随机游走的一个性质：反转轨道的增长。利用这个性质，有可能确定区间交换变换群的顺从性。另一方面，对这一财产的研究导致了一个更普遍的理论，这是计划发展。这个奖项反映了NSF的法定使命，并已被认为是值得支持的，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.