$\ell^{2}$-torsion of free-by-cyclic groups

Abstract

We provide an upper bound on the $\ell^{2}$-torsion of a free-by-cyclic group, $-ρ^{(2)}(\mathbb{F} \rtimes_Φ \mathbb{Z})$, in terms of a relative train-track representative for $Φ\in \mathrm{Aut}(\mathbb{F})$. Our result shares features with a theorem of Lück-Schick computing the $\ell^{2}$-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the $\ell^{2}$-torsion is determined by the exponential dynamics of the monodromy. In light of the result of Lück-Schick, a special case of our bound is analogous to the bound on the volume of a 3-manifold that fibers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima-McShane.