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

Abstract

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