Boundaries and subgroup structure

This project addresses a key problem in the field of geometric group theory: to understand the relation between geometric and algebraic structures and properties of a group. Research in this area has motivation coming from mathematical theory as well as from applications in cryptography and algorithms. In this project, the PI will investigate groups endowed with a geometric structure, and study situations when such a group can have subgroups that are geometrically ill-behaved, for example, having the property of being geometrically infinite. The PI will also engage in several projects aimed at broadening participation in mathematics by providing resources and creating an environment that supports and welcomes under-represented populations into mathematics. This will include giving expository lecture series, advocating for equity in publishing, organizing conferences and participating in the Tufts University Prison Initiative, seeking ways to improve their math training program for incarcerated and formerly incarcerated people. The PI will investigate the relation between three important topics in the theory of hyperbolic and relatively hyperbolic groups. The first is the boundary of a hyperbolic group. This is a canonical topological object that carries a lot, but not complete, group theoretical information about the group. The second is the quasi-isometry type of a group. Many properties are invariant under quasi-isometry, in particular the boundary. The third is the subgroup structure of groups whose finitely generated subgroups are not necessarily quasi-convex. Very little is known about this outside of hyperbolic 3-manifold groups, and the PI is pushing this to other classes of groups. 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.