Dynamics on free-by-cyclic groups

Dowdall, Spencer¹ ; Kapovich, Ilya² ; Leininger, Christopher J²

¹ Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
² Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA

Geometry & topology, Tome 19 (2015) no. 5, pp. 2801-2899.

Résumé

Given a free-by-cyclic group $G=F_N{⋊}_{\phi }\mathbb{Z}$ determined by any outer automorphism $\phi \in Out\left(F_N\right)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K\left(G,1\right)$ $2$–complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A}\subset H^1\left(X;ℝ\right)=Hom\left(G;ℝ\right)$ containing the homomorphism $u_0:G\to \mathbb{Z}$ having $ker\left(u_0\right)=F_N$, a homology class $ϵ\in H_1\left(X;ℝ\right)$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak{ℌ}:\mathcal{A}\to ℝ$ with the following properties. Given any primitive integral class $u\in \mathcal{A}$ there is a graph ${\Theta }_u\subset X$ such that:

Détail

DOI : 10.2140/gt.2015.19.2801

Classification : 20F65
Keywords: train track map, free-by-cyclic group, entropy

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