Rank gradient of small covers

Abstract

We prove that if $M \longrightarrow P$ is a small cover of a compact right-angled hyperbolic polyhedron $P$ then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient. As a corollary, if $\pi_1(M)$ is commensurable with the reflection group of $P$, then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient.