
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 is a small cover of a compact right-angled hyperbolic polyhedron then admits a cofinal tower of finite sheeted covers with positive rank gradient. As a corollary, if is commensurable with the reflection group of , then admits a cofinal tower of finite sheeted covers with positive rank gradient.
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