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

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.