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

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