Boundaries and Nonpositive Curvature

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The collection of symmetries of an object has a mathematical structure known as a group. Group theory has enjoyed a wide range of applications both in the physical world and the ideal mathematical universe. A particularly fruitful direction in the study of groups has been to consider symmetries of geometric spaces of nonpositive curvature and their associated boundaries. A geometric space is nonpositively curved if it is not "rounded" in any significant way. That is, it is either flat (such as the Euclidean plane), saddle-shaped, or some combination of the two. Given an infinite geometric object, one can attach a boundary at infinity in a way that allows us to view it from the outside in. This project centers on investigation of boundary theory in the context of nonpositively curved spaces. The results are expected to both expand the class of groups that are accessible through this theory and help define its limitations. The award includes support for graduate students involved in the project.The project focuses on structure-preserving automorphisms of products of hyperbolic spaces and CAT(0) cube complexes together with their associated boundaries. The investigator and collaborators will: (1) introduce and study irreducibly-acylindrical actions on products of hyperbolic spaces and examine persistence of known properties of acylindrically hyperbolic groups; (2) establish automorphism-equivariant relationships among three of the key natural boundaries associated to CAT(0) cube complexes: the Roller, simplicial, and Tits boundaries; and (3) examine the quality of convergence of a random walk on a CAT(0) cube complex, specifically to establish a Central Limit Theorem.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.