Exponential growth of torsion in Abelian coverings

TL;DR

This paper investigates how the torsion in homology groups of abelian covering spaces grows exponentially, especially in the context of link complements with certain Alexander polynomial properties.

Contribution

It proves exponential growth of torsion in abelian covers under specific conditions related to the Alexander polynomial's Mahler measure.

Findings

Torsion growth is exponential in the studied tower of covers.

Growth rate depends on the Mahler measure of the Alexander polynomial.

Results apply to towers converging to the maximal free abelian cover.

Abstract

We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler measure.