Dynamics on Free-by-cyclic Groups

Given a free-by-cyclie group G = FN oφ Z determined by a hyperbolic and fully irreducible outer automorphism φ ∈ Out(FN ), with u0 : G → Z the homomorphism having ker(u0) = FN , we construct a K(G, 1) 2–complex X called the folded mapping torus of φ, 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 u0, a homology class ∈ 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), (2) u( ) = χ(Θu), (3) Θu ⊂ X is a section of the semiflow and the first return map to Θu is an irreducible train track map representing φu ∈ Out(ker(u)) such that G = ker(u)oφu Z, (4) φu is hyperbolic and fully irreducible, (5) the logarithm of the stretch factor of φu is precisely H(u). In addition, we describe connections with the work of Button, Dunfield, and McMullen on the Alexander norm, work of Algom-Kfir and Rafi on stretch factors, as well as work of Gautero and Wang on section of semiflows.