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

TL;DR

This paper characterizes epimorphisms with finitely generated kernels from the mapping torus of polynomially growing free group automorphisms, describing all such structures and computing kernel ranks using hierarchical decompositions and Bieri-Neumann-Strebel invariants.

Contribution

It provides a complete classification of epimorphisms with finitely generated kernels for these groups and introduces methods to compute the Bieri-Neumann-Strebel invariant for graphs of groups.

Findings

Identifies all epimorphisms with finitely generated kernels from the mapping torus.

Computes the rank of kernels for these epimorphisms.

Describes the structure of the group as a mapping torus of a free group automorphism.

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.