Digraphs and cycle polynomials for free-by-cyclic groups

Abstract

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 $\varphi$ determines a free-by-cyclic group $\Gamma =F_n{⋊}_{\varphi }\mathbb{Z},$ and a homomorphism $\alpha \in H^1(\Gamma ;\mathbb{Z})$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, $\alpha$ has an open cone neighborhood $\mathcal{A}$ in $H^1(\Gamma ;\mathbb{R})$ 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 automorphism in $\mathcal{A}$.

41 pages, 20 figures

References in corpus (1)

• Mapping tori of small dilatation irreducible train-track maps