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

Abstract

We investigate Friedl and Lück’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 seminorm on the first cohomology of the group which is an analogue of the Thurston norm for $3$–manifold groups.

We prove that this Thurston seminorm is an upper bound for the Alexander seminorm defined by McMullen, as well as for the higher Alexander seminorms 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 $GL\left(n,\mathbb{Z}\right)$, 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.