Geometry & Topology

Dynamics on free-by-cyclic groups

Spencer Dowdall, Ilya Kapovich, and Christopher J Leininger

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Given a free-by-cyclic group G = FN φ 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; ) = Hom(G; ) containing the homomorphism u0: G having ker(u0) = FN, a homology class ϵ H1(X; ), and a continuous, convex, homogeneous of degree 1 function : A 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.
  4. The logarithm of the stretch factor of φu is precisely (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.

Article information

Geom. Topol., Volume 19, Number 5 (2015), 2801-2899.

Received: 6 June 2014
Revised: 30 December 2014
Accepted: 26 January 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

train track map free-by-cyclic group entropy


Dowdall, Spencer; Kapovich, Ilya; Leininger, Christopher J. Dynamics on free-by-cyclic groups. Geom. Topol. 19 (2015), no. 5, 2801--2899. doi:10.2140/gt.2015.19.2801.

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