Dynamics on free-by-cyclic groups

Abstract

Given a free-by-cyclic group G=FN⋊φℤ determined by any outer automorphism φ∈Out(FN) 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 A⊂H1(X;ℝ)=Hom(G;ℝ) containing the homomorphism u0:G→ℤ having ker(u0)=FN, a homology class ϵ∈H1(X;ℝ), and a continuous, convex, homogeneous of degree −1 function H:A→ℝ with the following properties. Given any primitive integral class u∈A there is a graph Θu⊂X such that 1.The inclusion Θu→X is π1–injective and π1(Θu)=ker(u). u(ϵ)=χ(Θu). Θu⊂X is a section of the semiflow and the first return map to Θu is an expanding irreducible train track map representing φu∈Out(ker(u)) such that G=ker(u)⋊φuℤ. The logarithm of the stretch factor of φu is precisely H(u). If φ was further assumed to be hyperbolic and fully irreducible then for every primitive integral u∈A the automorphism φu of ker(u) is also hyperbolic and fully irreducible.