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

We provide an upper bound on the $\ell^{2}$-torsion of a free-by-cyclic group, $-\rho^{(2)}(\mathbb{F} \rtimes_{\Phi} \mathbb{Z})$, in terms of a relative train-track representative for $\Phi \in \mathrm{Aut}(\mathbb{F})$. Our result shares features with a theorem of L\"uck-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 L\"uck-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.