  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.
