Dynamics and Rigidity for Discrete Subgroups of Lie Groups Beyond Lattices

Mathematical systems underlie today’s data encryption, navigation, and imaging technologies. Yet many of their most important behaviors occur in “infinite‐volume” settings that current theory cannot explain. This project develops new mathematical tools to understand the long-term dynamics and inherent rigidity of such systems, with an emphasis on groups that model negatively curved-space symmetries. By advancing the frontier of pure mathematics, the work directly fulfills the National Science Foundation’s mission “to promote the progress of science.” The results will be presented widely, including in a plenary lecture at the International Congress of Mathematicians (ICM 2026)—the world’s largest mathematics meeting—taking place July 23–30 2026 in Philadelphia, Pennsylvania, USA (https://www.mathunion.org/icm/icm-2026.) Sharing these results on a global stage accelerates discovery in the mathematical sciences and inspires the next generation of STEM talent. The project also incorporates research opportunities for graduate students and postdoctoral scholars. The project investigates dynamics on infinite-volume homogeneous spaces which are quotients of semisimple Lie groups by their discrete subgroups, with a focus on Kleinian groups and higher-rank Anosov subgroups.Goals are to (1) establish ergodicity, quantitative mixing, and orbit-closure criteria for diagonal and unipotent flows, (2) classify invariant measures in settings where finite-volume methods fail, and (3) derive sharp orbit-counting and equidistribution theorems via thermodynamic formalism. Methods blend homogeneous dynamics, Riemannian geometry, geometric group theory, Lie-theoretic representation theory, and spectral/transfer-operator techniques. Expected outcomes include new measure-rigidity theorems, effective counting asymptotics, and a unified framework connecting Kleinian, Anosov, and arithmetic lattice dynamics, laying groundwork for applications in geometry and number theory. 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.