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Digraphs and cycle polynomials for free-by-cyclic groups

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