Analytic Properties of Group Actions of Finitely Generated Groups

该项目的核心是基于PI最近开发的技术，旨在加深对群体及其行为的几个分析性质的理解。该研究的中心概念之一是适应性，它涉及到在群元素平移下不变的平均操作。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”有许多用途，多年来人们付出了相当大的努力，以表明各种群体都是顺从的。PI将研究一种新发展的群体行为性质：Liouville性质。该属性可以用来证明不可服从性，因为如果一个群体承认非liouville行为，那么该群体是不可服从的。PI已经开发了用于验证动作图的Liouville属性的工具。这些工具将强传递群作用的Liouville性质与加性组合联系起来。该项目用简化的方法描述了关于Liouville行动的几个问题，以及更雄心勃勃的问题和猜想。特别地，本文提出了一种关于Thompson群F的不服从性的方法，这是群论中一个广泛开放的问题。PI提供了一种区间交换转换组可调节性的方法。他们的行为是刘维尔式的，而且，他们完全是刘维尔式的。因此，不能用刘维尔性质来接近这些群的适性。出于这个原因，PI将研究作用随机游走的一个性质：反向轨道的增长。使用此属性有可能确定区间交换变换组的适应性。另一方面，对这一性质的研究导致了一种计划发展的更普遍的理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.