Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups

TL;DR

This paper explores the relationships between various norms and invariants, such as the Alexander and Thurston norms, for free-by-cyclic groups, revealing new bounds, computations, and equivalences that extend known results from 3-manifold groups.

Contribution

It establishes inequalities between the Thurston semi-norm and Alexander semi-norms for free-by-cyclic groups, and provides explicit methods to compute and relate the Bieri-Neumann-Strebel invariant with the universal L^2-torsion.

Findings

Thurston semi-norm bounds the Alexander semi-norm for these groups.

The Newton polytope of the L^2-torsion determines the BNS invariant.

In certain cases, the Alexander, Thurston, and higher Alexander norms coincide.

Abstract

We investigate Friedl-L\"uck's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb{Z}$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the universal $L^2$-torsion of a descending HNN extension of $F_2$ locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of $F_2$ has finitely many connected components. When the HNN extension is taken over $F_n$ along a polynomially growing automorphism with unipotent image in $GL(n, \mathbb{Z})$, we show that the Newton polytope of the universal $L^2$-torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.