Asymptotics of twisted Alexander polynomials and hyperbolic volume

Abstract

For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove the asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of Müller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials.

Analytic torsion, dynamical zeta functions, and the Fried conjecture

“…The flat vector bundle F 0 ⊗ F in Corollary 0.3 is of particular interest in study of hyperbolic volumes (see e.g. [BéDHP19]).…”

Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups

“…H. Goda and L. Bénard, J. Dubois, M. Heusener and J. Porti provided the asymptotic behavior of the higher-dimensional Reidemeister invariants with the descriptions in terms of the twisted Alexander invariants. These descriptions are given for hyperbolic knot complements in [God17] and cusped hyperbolic 3-manifolds in [BDHP19].…”