Limit Multiplicities and Betti numbers for congruence groups

We will discuss the growth of multiplicities of unitary representations in the space $L^{^2}(\Gamma_n\backslash G)$ and its relation to the geometry of $\Gamma_n\backslash G/K$ for a Lie group $G$ and a sequence of arithmetic subgroups $\Gamma_n$, which may be a sequence of congruence subgroups of a given lattice, or a sequence of noncommensurable lattices. (Part of this is joint work with M. Ab\'ert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov and I. Samet.)