#### Volume 19, issue 5 (2015)

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Dynamics on free-by-cyclic groups

### Spencer Dowdall, Ilya Kapovich and Christopher J Leininger

Geometry & Topology 19 (2015) 2801–2899
##### Abstract

Given a free-by-cyclic group $G={F}_{N}{⋊}_{\phi }ℤ$ 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 $\mathsc{A}\subset {H}^{1}\left(X;ℝ\right)=Hom\left(G;ℝ\right)$ containing the homomorphism ${u}_{0}:G\to ℤ$ 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 $\mathfrak{ℌ}:\mathsc{A}\to ℝ$ with the following properties. Given any primitive integral class $u\in \mathsc{A}$ 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 $G=ker\left(u\right){⋊}_{{\phi }_{u}}ℤ$.
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 $u\in \mathsc{A}$ the automorphism ${\phi }_{u}$ of $ker\left(u\right)$ is also hyperbolic and fully irreducible.
##### Keywords
train track map, free-by-cyclic group, entropy
Primary: 20F65