# Examples of homological torsion and volume growth

## TL;DR

This paper constructs explicit examples of hyperbolic 3-manifolds with towers of covers, linking homological torsion growth to volume growth, and relates spanning tree entropy to hyperbolic polyhedron volume.

## Contribution

It provides explicit computations of homological torsion growth in hyperbolic 3-manifolds derived from abelian covers of links, connecting algebraic and geometric properties.

## Key findings

- Homological torsion growth is explicitly computed in terms of volume.
- Examples are derived from abelian covers of alternating links.
- Spanning tree entropy equals the volume of a hyperbolic polyhedron.

## Abstract

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular planar lattice is given by the volume of a hyperbolic polyhedron.