  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.
