In the main talk we analyze the BNS invariant for hierarchies of group splittings. Mapping tori of polynomially free group automorphisms admit particularly nice hierarchies of splittings over infinite cyclic groups, and we are able to give a complete description of their fibrations. For these groups, the BNS invariant is symmetric, which is similar to the case of the mapping torus of a surface automorphism. In contrast, the BNS invariant of the mapping torus of an exponentially growing free group automorphism need not be symmetric.

This is joint work with Gilbert Levitt.