Georgia Institute of Technology College of Sciences

It is a natural question to ask whether one can deduce topological properties of a finite--volume three--manifold from its Riemannian invariants such as volume and systole. In all generality this is impossible, for example a given manifold has sequences of finite covers with either linear or sub-linear growth. However under a geometric assumption, which is satisfied for example by some naturally defined sequences of arithmetic manifolds, one can prove results on the asymptotics of the first integral homology. I will try to explain these results in the compact case (this is part of a joint work with M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov and I. Samet) and time permitting I will discuss their extension to manifolds with cusps such as hyperbolic knot complements.