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Let  G  be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from  G  to 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.

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