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

该项目制定了一个程序来研究一般群体的随机游走和渗透的几何和动力学方面，建立它们的代数结构、几何和动力学之间的联系。在过去的半个世纪里，人们对欧几里得空间中的格进行了经典的研究，并对一般格进行了研究，但本项目在丰富的新环境中探索了这些概念，他们的研究将应用于几何学、拓扑学和几何群论中的广泛问题。该项目还包括培养博士生和继续指导本科生。利用动力学和代数方法的结合，PI将研究具有算术特征的半单李群的子群结构（即密集通约子），以及与之密切相关的乘积的不可约子群的概念。其次，该项目将利用谱间隙方法的最新突破来研究随机漫步的渐近行为，以建立绝对连续性的一般性和进一步的正则性。在另一个方向上，该项目将研究无限群的可拓强度，推广经典（有限）可拓图，通过使用随机游走并将其与底层群的代数结构联系起来。最后，本课题将从几何群论的角度研究有限生成群的Cayley图上的渗流，特别是双曲群上渗流簇的度量畸变和边界理论。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 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.