Rank gradient of small covers

TL;DR

This paper proves that small covers of certain hyperbolic polyhedra have cofinal towers of finite covers with positive rank gradient, revealing new properties of their fundamental groups and their towers.

Contribution

It establishes the existence of cofinal towers with positive rank gradient for small covers of hyperbolic polyhedra and related reflection groups, advancing understanding of their group structures.

Findings

Existence of cofinal towers with positive rank gradient for small covers.

Extension of results to groups commensurable with reflection groups.

Implications for the structure of fundamental groups of these manifolds.

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.