Fun with finite covers of 3-manifolds: connections between topology, geometry, and arithmetic

Following the revolutionary work of Thurston and Perelman, the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In turn, hyperbolic geometry opens the door to applying tools from number theory, specifically automorphic forms, to what might seem like purely topological questions. After a passing wave at the recent breakthrough results of Agol, I will focus on exciting new questions about the geometric and arithmetic meaning of torsion in the homology of finite covers of hyperbolic 3-manifolds, motivated by the recent work of Bergeron, Venkatesh, Le, and others. I will include some of my own results in this area that are joint work with F. Calegari and J. Brock.