Abstracts for Seminar Homology Growth in Geometry and Arithmetic

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.

In this talk we concentrate on L^{^2}-invariants and questions about approximation by finite-dimensional invariants
and about homological growth. We present a series of open interesting questions which are in the focus
of actual research, give a status report and try to outline some ideas for a possible proof. If time allows,
we will discuss also Mahler measures and Lehmer's problem.

Let G be a group. One can use finite order automorphisms of G to give lower bounds for the Betti numbers.
For coefficients in characteristic zero this can be done using Lefschetz numbers. For coefficients in positive characteristic this can be achieved using Smith theory. In this talk I explain how to use these techniques to provide simple lower bounds for the growth of Betti numbers in p-adic analytic towers of S-arithmetic groups.