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

Abstract

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

23 pages; v2: incorporated comments from referee, to appear in The Quarterly Journal of Mathematics

References in corpus (1)

• Twisting $L^2$-invariants with finite-dimensional representations

Cited by in corpus (1)

• Cubulating mapping tori of some polynomial growth free group automorphisms