Mapping tori of free group automorphisms, and the Bieri-Neumann-Strebel invariant of graphs of groups

Abstract

Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing $G$ as the mapping torus of a free group automorphism. This is similar to the case for 3--manifold groups, and different from the case of mapping tori of exponentially growing free group automorphisms. The proof uses a hierarchical decomposition of $G$ and requires determining the Bieri-Neumann-Strebel invariant of the fundamental group of certain graphs of groups.

21 pages

References in corpus (2)

• The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth
• Notes on the Sigma invariants

Cited by in corpus (9)

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• Graphs and complexes of lattices
• On the Bieri-Neumann-Strebel-Renz ${\Sigma }^1$-invariant of even Artin groups
• Graphical splittings of Artin kernels
• Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups
• Cubulating mapping tori of some polynomial growth free group automorphisms
• Profinite rigidity for free-by-cyclic groups with centre