Profinite rigidity for twisted Alexander polynomials

TL;DR

This paper proves a profinite rigidity theorem for twisted Alexander polynomials, explores torsion growth in 3-manifold covers, and examines examples related to knot groups and hyperbolic volumes.

Contribution

It introduces a new profinite rigidity theorem for twisted Alexander polynomials and connects torsion growth with Mahler measures in 3-manifold covers.

Findings

Proved a profinite rigidity theorem for twisted Alexander polynomials.

Derived torsion growth formulas using Mahler measures.

Analyzed examples from knot groups and hyperbolic volume remarks.

Abstract

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley's parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.