Geometry, Arithmeticity, and Random Walks on Discrete and Dense Subgroups of Lie Groups

李群的离散和稠密子群上的几何、算术和随机游走

• 批准号: 2203867
• 负责人: Wouter Van Limbeek
• 金额: $ 17.5万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203867&HistoricalAwards=false
• 关键词: Geometry; Arithmeticity; Random; Walks; Discrete

The project lays out a program to study geometric and dynamical aspects of random walks and percolation on general groups, building connections between their algebraic structure, geometry, and dynamics. Classically this has been studied for lattices in Euclidean space, and over the last half century, for general lattices, but this project explores these notions in rich, new settings, where their study will have applications to a wide-ranging collection of problems in geometry, topology, and geometric group theory. The project also includes training of PhD students and continued mentoring of undergraduate students.Using a combination of dynamical and algebraic methods, the PI will study the structure of subgroups of semisimple Lie groups with arithmetic features (namely dense commensurator), as well as the closely related notion of irreducible subgroups of products. Secondly, the project will study the asymptotic behavior of random walks using recent breakthroughs in spectral gap methods to establish genericity of absolute continuity and further regularity properties. In a different direction, the project will study expansion strength of infinite groups, generalizing the classical (finite) expander graphs, by using random walks and relating this to the algebraic structure of the underlying group. Finally, the project will study percolation on Cayley graphs of finitely generated groups from the point of view of geometric group theory, especially metric distortion and boundary theory of percolation clusters on hyperbolic groups.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.

该项目提出了一个计划，以研究随机步行的几何和动态方面，并在一般群体上进行渗透，建立其代数结构，几何和动态之间的联系。从经典的角度来看，这是在欧几里得空间中以及过去半个世纪的一般晶格中研究的，但是该项目在丰富的新环境中探讨了这些概念，他们的研究将在几何，拓扑，拓扑和几何组理论中应用大量的问题收集。 该项目还包括对博士生的培训以及继续指导本科生。使用动力学和代数方法的组合，PI将研究具有算术特征（即密集的致密者）的半圣母谎言组的亚组的结构，以及不可避免的产品的不可避免的概念。其次，该项目将使用频谱差距方法中的最新突破来研究随机步行的渐近行为，以建立绝对连续性和进一步的规律性属性的通用性。在不同的方向上，该项目将通过使用随机步行并将其与基础组的代数结构联系起来，研究无限基团的扩展强度，从而概括了经典的（有限）扩展器图。最后，该项目将从几何群体理论的角度研究有限生成的小组的Cayley图表，尤其是在双曲线组上渗透群集的度量失真和边界理论。这项奖项反映了NSF的法定任务，并认为通过基金会的知识分子优点和广泛的影响来评估NSF的法定任务。