#### Volume 19, issue 2 (2015)

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### Yael Algom-Kfir, Eriko Hironaka and Kasra Rafi

Geometry & Topology 19 (2015) 1111–1154
##### 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 $\Gamma ={F}_{n}{⋊}_{\varphi }ℤ$ and a homomorphism $\alpha \in {H}^{1}\left(\Gamma ;ℤ\right)$. By work of Neumann, Bieri, Neumann and Strebel, and Dowdall, Kapovich and Leininger, $\alpha$ has an open cone neighborhood $\mathsc{A}$ 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 $\mathsc{A}$.

##### Keywords
fibrations, free-by-cyclic groups, generalizations of the Teichmüller polynomial
Primary: 57M20