Dynamics on free-by-cyclic groups

Abstract

Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ 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 $\epsilon \in H_1(X;\mathbb{R})$, 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 $\Theta_u \subset X$ such that: (1) the inclusion $\Theta_u \to X$ is $\pi_1$-injective and $\pi_1(\Theta_u) = \mathrm{ker}(u)$, (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 $\varphi_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{\varphi_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $\varphi_u$ is precisely $\mathfrak H(u)$, (5) if $\varphi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $\varphi_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.Comment: 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 & Topolog

The Bieri–Neumann–Strebel invariants via Newton polytopes

“…Let us note that understanding the BNS invariants of free-by-cyclic groups is related to a large body of current research, see [DKL1,DKL2,DKL4] by Dowdall-I. Kapovich-Leininger and [AKHR] by Algom-Kfir-Hironaka-Rafi.…”

ℓ2-TORSION OF FREE-BY-CYCLIC GROUPS

“…Example 4.4. We will construct the Jacobians J 0 (f ) and J 1 (f ) for the "Running Example" f : Γ → Γ of [9] (Example 2.2) and [7] (Example 3.3). The graph Γ is shown in Figure 2 (we have reversed the orientation on some edges).…”

Unbounded asymmetry of stretch factors

“…The proof will appeal to a construction and analysis carried out in [DKL1] and [DKL2]. To that end, let F 3 = a, b, c and consider the element ϕ ∈ Aut(F 3 ) defined by…”

“…In [DKL1], we construct a cone A ⊂ H 1 (G; R) containing u 0 with the property that every other primitive integral element u ∈ A has kernel ker(u) a finitely generated free group. The action of u(G) = Z on ker(u) is generated by a monodromy automorphism ϕ u ∈ Aut(ker(u)) determining an expression of G as a semidirect product G ∼ = ker(u) ⋊ ϕu Z with associated homomorphism u.…”

“…The action of u(G) = Z on ker(u) is generated by a monodromy automorphism ϕ u ∈ Aut(ker(u)) determining an expression of G as a semidirect product G ∼ = ker(u) ⋊ ϕu Z with associated homomorphism u. One of the main results of [DKL1] is that all such ϕ u are fully irreducible.…”