Digraphs and cycle polynomials for free-by-cyclic groups

Letфε2 Out.F_n) be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism ф determines a freeby-cyclic group Г=F_n⋊_ф Z and a homomorphism αεH¹(Г;Z). By work of Neumann, Bieri, Neumann and Strebel, and Dowdall, Kapovich and Leininger, α has an open cone neighborhood A in H¹(Г;∞) whose integral points correspond to other fibrations of Г whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen’s Teichmüller polynomial that computes the dilatations of all outer automorphisms in A.