The leading coefficient of the $L^2$-Alexander torsion

Abstract

We give upper and lower bounds on the leading coefficients of the $L^2$-Alexander torsions of a $3$-manifold $M$ in terms of hyperbolic volumes and of relative $L^2$-torsions of sutured manifolds obtained by cutting $M$ along certain surfaces. We prove that for numerous families of knot exteriors the lower and upper bounds are equal, notably for exteriors of 2-bridge knots. In particular we compute the leading coefficient explicitly for 2-bridge knots.

v2, final version : 33 pages, 5 figures. we rewrote parts of the paper, we added details to the proofs and we added Figure 3 for clarity, but the mathematical content is globally unchanged. Accepted for publication at the Annales de l'Institut Fourier