LEAPS-MPS: Cubulation and Property (T) in Random Groups

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Groups are mathematical structures that describe the symmetry of a geometric object. They are used throughout the sciences: in the study of crystalline structures, molecular symmetry, the Standard Model of particle physics, public-key encryption systems, and more. This project aims to understand groups in their own right; instead of starting with an object and calculating its symmetry group as a chemist or physicist might, a mathematician can start with an abstract group and then study the space(s) whose symmetries it could describe. A natural question to ask is: What properties does a `typical’ group satisfy? This project will focus on understanding these `typical’ properties of groups by introducing and investigating a new model of random groups. The project will also support undergraduate projects and the PI’s ongoing leadership and organizational efforts to promote inclusivity and connections for undergraduate women and other under-represented groups within mathematics, such as the student group Gender Minorities in Math/Stats (GeMMs), mentor/mentee programs, book clubs and other community building activities at Carlton College.Gromov random groups have been a rich source of examples in geometric group theory. The first branch of this project introduces and explores properties of a new model of random quotients of free products of groups, which is combinatorially related to Gromov’s model. In collaboration with Einstein, Krishna, Ng, and Steenbock the PI will investigate (relative) cubulation in this setting. The PI will also explore Property (T) for these groups. In another direction, the PI will work with undergraduates to further our understanding of cubulation in Gromov random groups; in particular they will increase the known bound for cocompact actions on CAT(0) cube complexes.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.