Dynamics on free-by-cyclic groups

Abstract

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:

1. The inclusion ${\Theta }_u\to X$ is ${\pi }_1$–injective and ${\pi }_1\left({\Theta }_u\right)=ker\left(u\right)$.

2. $u\left(ϵ\right)=\chi \left({\Theta }_u\right)$.

3. ${\Theta }_u\subset X$ is a section of the semiflow and the first return map to ${\Theta }_u$ is an expanding irreducible train track map representing ${\phi }_u\in Out\left(ker\left(u\right)\right)$ such that $G=ker\left(u\right){⋊}_{{\phi }_u}\mathbb{Z}$.

4. The logarithm of the stretch factor of ${\phi }_u$ is precisely $\mathfrak{ℌ}\left(u\right)$.

5. If $\phi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism ${\phi }_u$ of $ker\left(u\right)$ is also hyperbolic and fully irreducible.