Rigidity and bounded cohomology of groups acting on nonpositively curved spaces

The PI will investigate collections of transformations of objects or spaces, known as transformation groups, which arise ubiquitously across mathematics, science, and engineering. The project will focus on understanding the fundamental dynamical, topological, and geometric properties of infinite transformation groups that act in a distance-preserving manner on spaces of nonpositive curvature, such as Euclidean and hyperbolic spaces. A key component of the project involves studying and computing topological invariants of these groups, including their ordinary and bounded cohomology. The PI and collaborators have previously established rigidity of such group actions in certain contexts, building on this foundation, the current project will tackle several challenging open conjectures using recent advancements in these techniques. The award will also support the PI to improve mathematics and STEM education at all levels. The PI will conduct three main projects. The first is to investigate the subtle interplay between the geometry of nonpositively curved closed Riemannian manifolds and the bounded cohomology of their fundamental groups. For instance, a conjecture of M. Gromov asks whether strictly negative Ricci curvature suffices to guarantee positivity of the simplicial volume in such manifolds. The second is to examine the implications of the "natural flow," a tool recently introduced by the PI and collaborators, for understanding the relationship between the (co)homological dimension of a discrete group acting on a Hadamard space and its critical exponent. The final project will deepen our understanding of local and global rigidity phenomena. In particular, the PI will extend recent work with collaborators on the local rigidity of higher rank cocompact lattices to the nonuniform setting. The PI also will continue advancing the hyperbolic rank rigidity program, building on foundational contributions by W. Ballmann, K. Burns, P. Eberlein, U. Hamenstädt, R. Spatzier, and others. 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.