NSF Award
2204339
As long as mathematics has been studied, people have sought to understand the relationship between geometry and symmetry. The Euclidean plane is most familiar, closely followed by the sphere. It has long been known that one cannot periodically tile the plane using the same arrangement of shapes as one would the sphere. This can be understood mathematically through the geometric notion of curvature: the plane is flat while the sphere is positively curved. This project concerns the vast universe of non-Euclidean geometries with non-positive curvature. The PI will investigate asymptotic invariants and rigidity phenomena in this setting, while supporting student involvement and broadened participation in mathematics via mentoring and community outreach. <br/><br/><br/> This research concerns finitely generated groups and their large-scale geometry. The first project investigates graphical discreteness, a notion that unifies two distinct programs of study: rigidity phenomena and classifying lattice envelopes. The former has been a central problem in geometric group theory, while the latter was initiated with Mostow--Prasad Rigidity, which characterized Lie group envelopes of hyperbolic manifold groups. The PI will consider a diverse family of examples, including hyperbolic groups with Menger curve boundary and groups that split as graphs of groups. The second project focuses on hyperbolic groups with Menger compacta visual boundaries and will build techniques to study the quasi-conformal structures on these spaces. The third project aims to study relatively hyperbolic groups and their boundaries via analytic methods and quasi-conformal geometry.<br/><br/>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.
