CAREER: Rigidity of Group Actions on Manifolds

Overview Group actions on various spaces arise naturally in many areas of mathematics. The study of such actions involves questions and techniques from geometric group theory, geometry, topology, representation theory, and dynamical systems. The proposal details a research program that investigates various rigidity phenomena for nonlinear actions of discrete groups on manifolds. Two related directions of research are discussed: (1) rigidity of actions by lattices in higher-rank Lie groups and the Zimmer program, and (2) rigidity and classification of orbit closures and stationary measures for actions satisfying certain dynamical hypotheses. Certain discrete groups such as SL(n, Z)for n ≥ 3 (and more general higher-rank lattices) are known to exhibit certain rigidity properties with respect to linear representations. The Zimmer program is a collection of conjectures and questions which roughly aims to extend known rigidity results for linear representations to actions by such groups on manifolds. These conjectures suggest that all actions of such groups are expected to be modifications of standard algebraic actions. In particular, such groups are not expected to act (non-trivially) on low-dimensional manifolds. In contrast, the group of integers Z or the free group on 2 generators F2 act (in non-algebraic ways) on any manifold. For certain families of homogeneous and affine actions, a number of recent extremely influential results establish that certain dynamically defined objects—namely orbit closures and invariant or stationary measures—may be classified by showing that all such objects are smooth or homogeneous. In contrast, for actions of one-parameter groups, Z-actions defined by a diffeomorphism f : M → M often have Cantor sets as orbit closures and singular invariant measures. Intellectual Merit The proposal outlines a program that would establish a number of conjectures within the Zimmer program. Techniques proposed by the PI to approach these conjectures involve a number of modern tools including recent results in rigidity theory for nonlinear actions by higher-rank abelian groups. The proposal also presents a program that would extend classification and rigidity results for orbit closures and invariant measures obtained in affine or homogeneous settings to certain inhomogeneous settings. These projects would combine techniques from nonlinear smooth dynamics and ergodic theory with tools developed in the rigidity theory for homogeneous dynamics. These techniques would apply to families of actions such as actions on character varieties which are of interest to mathematicians in various fields. The proposed projects interface between many areas of mathematics including smooth dynamics and ergodic theory, Lie theory, geometry, homogeneous dynamics, algebraic groups, and representation theory. The proposal has potential to expose recently developed tools in these areas to wider audiences and to further encourage exchange of ideas between different areas of mathematics. Broader Impacts The PI will complete a number of outreach and educational projects including the following: (1) Outreach to high school students through teaching month-long mathematics enrichment courses with the University of Chicago’s Young Scholars Program. (2) Organizing a workshop for graduate students and postdocs on recent results related to rigidity and flexibility in smooth dynamical systems and group actions. (3) Developing a course for undergraduates at the University of Chicago on dynamical systems. (4) Writing a monograph on dynamics and ergodic theory fo