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Dynamics on free-by-cyclic groups. (English) Zbl 1364.20026
Summary: 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 W. P. Thurston [Mem. Am. Math. Soc. 339, 99–130 (1986; Zbl 0585.57006); Bull. Am. Math. Soc., New Ser. 19, No. 2, 417–431 (1988; Zbl 0674.57008)] and D. Fried [Comment. Math. Helv. 57, 237–259 (1982; Zbl 0503.58026); Topology 21, 353–371 (1982; Zbl 0594.58041)] 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 \(\mathrm{ker}(u_0) = F_N\), a homology class \(\epsilon \in H_1(X;\mathbb R)\), and a continuous, convex, homogeneous of degree \(-1\) function \(\mathfrak{H}:\mathcal{A} \to \mathbb R\) 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.

MSC:
20F65 Geometric group theory
57M07 Topological methods in group theory
57M20 Two-dimensional complexes (manifolds) (MSC2010)
20E36 Automorphisms of infinite groups
37B40 Topological entropy
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