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

Abstract

We investigate Friedl-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 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.

36 pages; to appear in Geometry & Topology

References in corpus (1)

• Universal $L^2$-torsion, polytopes and applications to $3$-manifolds

Cited by in corpus (5)

• The Bieri-Neumann-Strebel invariants via Newton polytopes
• $L^2$-Euler characteristics and the Thurston norm
• On the Bieri-Neumann-Strebel-Renz ${\Sigma }^1$-invariant of even Artin groups
• Survey on L^2-invariants and 3-manifolds
• Handlebody Bundles and Polytopes