Rank gradient in co-final towers of certain Kleinian groups

TL;DR

This paper investigates the rank gradient behavior in towers of covers of certain hyperbolic 3-manifolds, demonstrating the existence of towers with positive rank gradient under specific group conditions.

Contribution

It establishes the existence of co-final towers with positive rank gradient for fundamental groups of certain hyperbolic 3-manifolds, extending understanding of their subgroup structures.

Findings

Existence of co-final towers with positive rank gradient

Manifolds with known towers of zero rank gradient also have towers with positive gradient

Conditions on the fundamental group relate to reflection groups of polyhedra

Abstract

We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has finite index in the reflection group of a right-angled ideal polyhedra in $\mathbb{H}^3$ then it has a co-final tower of finite sheeted covers with positive rank gradient. The manifolds we provide are also known to have co-final towers of covers with zero rank gradient.