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

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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.