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

##### External Resource

http://dx.doi.org/10.2140/gt.2018.22.2647

(Publisher version)

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##### Fulltext (public)

arXiv:1605.09067.pdf

(Preprint), 436KB

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##### Citation

Funke, F., & Kielak, D. (2018). Alexander and Thurston norms, and the Bieri-Neumann-Strebel
invariants for free-by-cyclic groups.* Geometry & Topology,* *22*(5),
2647-2696. doi:10.2140/gt.2018.22.2647.

Cite as: https://hdl.handle.net/21.11116/0000-0003-ACB7-D

##### 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 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 semi-norm is an upper bound

for the Alexander semi-norm 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 $GL(n,\mathbb{Z})$, 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.

$F_n$-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 semi-norm is an upper bound

for the Alexander semi-norm 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 $GL(n,\mathbb{Z})$, 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.