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

Florian Funke and Dawid Kielak

Geometry & Topology 22 (2018) 2647–2696

We investigate Friedl and Lück’s universal L2–torsion for descending HNN extensions of finitely generated free groups, and so in particular for Fn-by- 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 L2–torsion of a descending HNN extension of F2 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 F2 has finitely many connected components.

When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in GL(n, ), we show that the Newton polytope of the universal L2–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.

free-by-cyclic groups, ascending HNN extensions of free groups, BNS invariants, Thurston norm, Alexander norm
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 16S85, 20E06
Received: 31 May 2016
Revised: 26 October 2017
Accepted: 14 January 2018
Published: 1 June 2018
Proposed: Jean-Pierre Otal
Seconded: Martin Bridson, Walter Neumann
Florian Funke
Mathematisches Institut
Universität Bonn
Dawid Kielak
Fakultät für Mathematik
Universität Bielefeld