Asymptotics of twisted Alexander polynomials and hyperbolic volume

Joan Porti Pique, Michael Heusener, Jérôme Dubois, Léo Bénard

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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.
Original languageEnglish
Pages (from-to)1155-1207
JournalIndiana University Mathematics Journal
Volume71
Issue number3
Publication statusPublished - 2022

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