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Description

Given a hyperbolic 3-manifold with cusps, we consider the composition of a lift of its holonomy in \(\mathrm{SL}(2,\mathbb{C})\) with the irreducible representation in $\mathrm{SL}(n,\mathbb{C})$, that yields a twisted Alexander polynomial $A_n(t)$, for each natural $n$. We prove that, for a complex number $z$ with norm one $\log|A_n(z)|/n^2$ converges to the hyperbolic volume of the manifold divided by $4 \pi$, as $n\to\infty$. This generalizes and uses a theorem of W.Mueller for closed manifolds on analytic torsion. This is joint work with L.Bénard, J.Dubois and M.Heusener.