Geometry and Topology Seminar: "Arithmetic groups, thin groups, and commensurators"

Abstract: Arithmetic groups are an important class of lattices in Lie groups which are of interest from a dynamical, geometric, and number theoretic perspective. These groups were characterized among lattices in a purely intrinsically algebraic way, by a famous result of Margulis. I will survey some of the ideas surrounding arithmetic groups and Margulis' theorem, and then move on to a discussion of thin groups. Thin groups are certain discrete subgroups of Lie groups which occur naturally in many contexts in mathematics, from number theory and spectral theory to quantum computing. Thin groups have much less structure than lattices, though they seem to follow some organizational principles analogous to Margulis' theorem. I will survey some recent results in this direction.

Host: Michael Landry