Arithmetic hyperbolic n-manifolds are characterized by being infinite index in their commensurator. This implies that an arithmetic hyperbolic n-manifold either contains no totally geodesic hypersurfaces or they are everywhere dense. Reid and (independently) McMullen asked whether having infinitely many totally geodesic hypersurfaces conversely implies arithmeticity. I will discuss work with Bader, Fisher, and Miller that answers this question in the positive, focusing on the case of hyperbolic 3-manifolds. Our main tool is a superrigidity theorem for certain representations of fundamental groups of hyperbolic manifolds. Speaker(s): Matthew Stover (Temple U)