Dynamics on free-by-cyclic groups

Original · EN

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 → 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), (2) u(ε) = χ(Θᵤ), (3) 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) ᵩᵤ Z, (4) the logarithm of the stretch factor of φᵤ is precisely H(u), (5) 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.