Digraphs and cycle polynomials for free-by-cyclic groups

Abstract

Let $\varphi \in Out\left(F_n\right)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $\varphi$ determines a free-by-cyclic group $\Gamma =F_n{⋊}_{\varphi }\mathbb{Z}$ and a homomorphism $\alpha \in H^1\left(\Gamma ;\mathbb{Z}\right)$. By work of Neumann, Bieri, Neumann and Strebel, and Dowdall, Kapovich and Leininger, $\alpha$ has an open cone neighborhood $\mathcal{A}$ in $H^1\left(\Gamma ;ℝ\right)$ whose integral points correspond to other fibrations of $\Gamma$ 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 $\mathcal{A}$.