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

TL;DR

This paper establishes an upper bound on the $\,\ell^{2}$-torsion of free-by-cyclic groups using relative train-track representatives, linking algebraic invariants to the dynamics of automorphisms, similar to 3-manifold fiberings.

Contribution

It introduces a novel upper bound on $\,\ell^{2}$-torsion for free-by-cyclic groups based on monodromy dynamics, extending previous geometric results.

Findings

Upper bound on $\,\ell^{2}$-torsion in terms of automorphism dynamics

Connection between algebraic invariants and exponential dynamics

Analogy with volume bounds in 3-manifold fiberings

Abstract

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.