Digraphs and cycle polynomials for free-by-cyclic groups

Let 2 Out.Fn/ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism determines a freeby-cyclic group AD Fn A Z and a homomorphism 2 H 1 .AIZ/. By work of Neumann, Bieri, Neumann and Strebel, and Dowdall, Kapovich and Leininger, has an open cone neighborhood A in H 1 .AIR/ whose integral points correspond to other fibrations of A whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen’s Teichmuller polynomial that computes the dilatations of all outer automorphisms in A. 57M20