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

该项目设计了一个程序，研究一般群上随机行走和渗流的几何和动力学方面，在它们的代数结构、几何和动力学之间建立联系。经典地，这已经被研究在欧几里德空间中的格，并且在过去的半个世纪中，对于一般的格，但是这个项目在丰富的、新的环境中探索这些概念，他们的研究将应用于几何、拓扑和几何群论中的广泛的问题集合。该项目还包括对博士生的培训和对本科生的继续指导。PI将结合使用动力学和代数方法，研究具有算术特征(即稠密公约子)的半单李群的子群的结构，以及与乘积的不可约子群密切相关的概念。其次，该项目将利用谱间隙方法的最新突破来研究随机游动的渐近行为，以建立绝对连续性的一般性和进一步的正则性。在不同的方向上，该项目将研究无限群的扩张强度，通过使用随机游动并将其与基础群的代数结构相联系来推广经典的(有限)扩张图。最后，该项目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扭曲和边界理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.