Mod-$p$ cohomology growth in $p$-adic analytic towers of 3-manifolds

Abstract

Let $M$ be a compact 3-manifold with infinite fundamental group $\Gamma$. Given a homomorphism from $\Gamma$ to a $p$-adic analytic group $G$ with dense image, we describe the possible mod-$p$ homology growth of covers $M_n$ of $M$ determined by the congruence subgroups $G_n$. If $d$ = dim($G$) > 3, this growth is always non-trivial, growing at least as fast as Vol($M_n$)${}^{(d-1)/d}$.