Digraphs and cycle polynomials for free-by-cyclic groups

Algom-Kfir, Yael¹ ; Hironaka, Eriko² ; Rafi, Kasra³

¹ Mathematics Department, University of Haifa, Mount Carmel, Haifa 31905, Israel
² Department of Mathematics, Florida State University, 208 Love Building, 1017 Academic Way, Tallahassee, FL 32301-4510, USA
³ Department of Mathematics, University of Toronto, Room 6290, 40 George Street, Toronto ON M5S 2E4, Canada

Geometry & topology, Tome 19 (2015) no. 2, pp. 1111-1154.

Résumé

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 $\mathcal{A}$.

Détail

DOI : 10.2140/gt.2015.19.1111

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

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