Profinite rigidity for twisted Alexander polynomials (Q2121381)

The author considers and gives an answer for a question about profinite rigidity of twisted Alexander polynomials which can be stated roughly as follows: given a discrete group \(\pi\) and a representation \(\rho\) of that group, a profinite completion of \(\pi\) induces a representation \(\hat{\rho}\), to what extent does the isomorphism class of \(\hat{\rho}\) determine the twisted Alexander polynomial associated with \(\rho\)? From recently announced results for hyperbolic knots relating twisted Alexander polynomials to volumes, the author deduces that under certain assumptions for the holonomy representation \(\rho\) the isomorphism class of \(\hat{\rho}\) determines the volume of the knot. In a related setting the author gives a generalization of Fox's formula, relating growth of the torsion part of the first twisted homology of cyclic covers of a knot complement to a Mahler measure of the twisted Alexander polynomial. The results are demonstrated in several cases of two-bridge knots.