Dynamics on free-by-cyclic groups

Abstract

Given a free-by-cyclic group $G=F_N{⋊}_{\phi }\mathbb{Z}$ determined by any outer automorphism $\phi \in \mathrm{Out}(F_N)$ 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 $u_0:G\to \mathbb{Z}$ having $\ker (u_0)=F_N$, a homology class $\epsilon \in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak{H}:\mathcal{A}\to \mathbb{R}$ 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 ${\pi }_1({\Theta }_u)=\ker (u)$, (2) $u(\epsilon )=\chi ({\Theta }_u)$, (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 ${\phi }_u\in \mathrm{Out}(\ker (u))$ such that $G=\ker (u){⋊}_{{\phi }_u}\mathbb{Z}$, (4) the logarithm of the stretch factor of ${\phi }_u$ is precisely $\mathfrak{H}(u)$, (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 $\ker (u)$ is also hyperbolic and fully irreducible.

v7: Minor organizational and stylistic changes incorporating referee's suggestions. Notably, section 6.3 in v6 has been moved to section 4.5 in v7. 67 pages, 13 figures. Final version; accepted for publication in Geometry & Topology

References in corpus (3)

• On hyperbolicity of free splitting and free factor complexes
• Algorithmic detectability of iwip automorphisms
• Unbounded asymmetry of stretch factors

Cited by in corpus (7)

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• North-South dynamics of hyperbolic free group automorphisms on the space of currents
• Canonical measures on metric graphs and a Kazhdan's theorem
• Irreducible Nonsurjective Endomorphisms of $F_n$ are Hyperbolic
• Counting conjugacy classes of fully irreducibles: double exponential growth
• Handlebody Bundles and Polytopes
• Cubulating a free-product-by-cyclic group