Lower bounds for the growth of Betti numbers of arithmetic groups

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.