CAREER: Large scale geometry and negative curvature

The symmetries of a space and its relation to the underlying geometric structure of the space has led researchers to deep insights into the connections between algebraic and geometric structures. This project focuses on hyperbolic spaces and the coarse geometry their structure induces on their associated symmetry groups. The project activity also includes initiatives in teaching and mentoring mathematics students at all levels, both in the department and the region. The funded research has three main goals: to generalize notions of coarse negative curvature, particularly acylindrical hyperbolicity, to a broad class of topological groups; to establish a strong stability result for quotients of hierarchically hyperbolic groups, a class of groups which can be completely described by their actions on hyperbolic metric spaces; and to extend combinatorial methods from cubical groups to the larger class of CAT(0) groups, with a focus on aspects of negative curvature and boundaries. Parts of this research program will be incorporated into student projects as a way to expand access to undergraduate research at the PI’s institution. 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.