On lower bounds for cohomology growth in p-adic analytic towers

TL;DR

This paper investigates the growth of mod-l Betti numbers in p-adic analytic towers of groups, providing methods to lift lower bounds from fixed point subgroups to the entire group, with applications to S-arithmetic groups.

Contribution

It introduces a new technique to transfer lower bounds of cohomology growth from fixed point subgroups to the whole group in p-adic towers, including rational coefficients.

Findings

Established lower bounds for cohomology growth in fixed point groups.

Extended results to cohomology with rational coefficients.

Applied the theory to S-arithmetic groups.

Abstract

Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed point group G^X to lower bounds for the growth in G. We give applications to S-arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.