Topological aspects of infinite group theory

One of the fundamental geometric insights of the last hundred years is that symmetry (broadly interpreted) governs a vast array of phenomena. Even many seemingly non-geometric objects are amenable to study via symmetry, giving them a sort of hidden geometry that can be used to study questions that cannot be answered otherwise. This project will focus on topological aspects of this symmetry. Here topology refers to the study of large-scale properties of spaces that are invariant under bending and stretching; for instance, the presence of high-dimensional holes. The PI and his PhD students will investigate an array of questions in this area. The PI will also continue his engagement in high school outreach, conference organization, and expository writing with the aim of broadening participation in mathematics at a variety of levels. The investigator will make advances in the study of the topology of mapping class groups, automorphism groups of free groups, and arithmetic groups with an eye toward building bridges between objects of study in geometric topology and geometric group theory on the one hand, and algebraic geometry and representation theory on the other. The proposed work has four main directions. In the first, the PI will extend results on stable homology to finite-index subgroups. In the second, the PI will make advances in the open question when are mapping class groups and automorphism groups of free groups commensurable to the fundamental group of a compact Kahler manifold, thus developing a novel connection between groups arising in geometric topology and in algebraic geometry. In the third, the PI will prove a version of representation stability for the homology of the Torelli group. Finally, the PI will clarify the relations between the unstable homology of arithmetic groups, and the Steinberg representation. 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.