Dynamics on free-by-cyclic groups

Given a free-by-cyclic group $G = F_N \rtimes_φ\mathbb{Z}$ determined by any outer automorphism $φ\in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-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 $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $ε\in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $Θ_u \subset X$ such that: (1) the inclusion $Θ_u \to X$ is $π_1$-injective and $π_1(Θ_u) = \mathrm{ker}(u)$, (2) $u(ε) = χ(Θ_u)$, (3) $Θ_u \subset X$ is a section of the semiflow and the first return map to $Θ_u$ is an expanding irreducible train track map representing $φ_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{φ_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $φ_u$ is precisely $\mathfrak H(u)$, (5) if $φ$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $φ_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.

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