Arithmeticity, superrigidity and totally geodesic submanifolds

I will discuss a recent result on finite volume hyperbolic manifolds of dimension at least 3. We show that any such manifold with infinitely many maximal closed totally geodesic submanifolds has arithmetic fundamental group. A closed totally geodesic manifold is maximal if it is not contained in another closed proper totally geodesic submanifold. So in particular a closed totally geodesic submanifold of codimension one is always maximal. This answers a question of Reid and McMullen. The proof deduces arithmeticity from a superrigidity theorem and makes key use of equidistribution results from homogeneous dynamics. If time permits, I will also discuss some other applications of our techniques and some related open questions. This is joint work with Uri Bader, Nick Miller and Matthew Stover.