CAREER: Virtual properties of groups in geometry and topology

This project supports research at the intersection of geometry, topology, and algebra as well as educational activities focused on local mathematical communities. The PI will investigate the relationship between the algebraic properties of groups and the geometric and topological properties of the spaces on which the groups act on. The educational activities of the project include a workshop for students and faculty in the upper Midwest region, a research seminar series bringing together geometers and topologists from nearby institutions, and supporting undergraduates from underrepresented groups through the Mathematics Project at Minnesota. The first research goal of the project is to study virtual properties of fundamental groups of hyperbolic manifolds, that is, properties of their finite-index subgroups since these correspond to finite-sheeted covers of the manifolds. The PI will consider the case of hyperbolic 3-manifolds as well as hyperbolic manifolds in higher dimensions with an emphasis on those constructed via arithmetic methods. The second research goal of the project is to explore the virtual properties of certain classes of groups which generalize and extend fundamental groups of hyperbolic manifolds. The PI will consider questions on the subgroup structure of these groups and their geometric actions. 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.