Dynamics on free-by-cyclic groups

Dowdall, Spencer; Kapovich, Ilya; Leininger, Christopher J

doi:10.2140/gt.2015.19.2801

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Dynamics on free-by-cyclic groups

Spencer Dowdall, Ilya Kapovich and Christopher J Leininger

Geometry & Topology 19 (2015) 2801–2899

DOI: 10.2140/gt.2015.19.2801

Abstract

Given a free-by-cyclic group $G = F_{N} ⋊_{φ} ℤ$ determined by any outer automorphism $φ \in Out (F_{N})$ 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 Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $A \subset H^{1} (X; ℝ) = Hom (G; ℝ)$ containing the homomorphism $u_{0} : G \to ℤ$ having $ker (u_{0}) = F_{N}$ , a homology class $ϵ \in H_{1} (X; ℝ)$ , and a continuous, convex, homogeneous of degree $- 1$ function $ℌ : A \to ℝ$ with the following properties. Given any primitive integral class $u \in A$ there is a graph $Θ_{u} \subset X$ such that:

The inclusion $Θ_{u} \to X$ is $π_{1}$ –injective and $π_{1} (Θ_{u}) = ker (u)$ .
$u (ϵ) = χ (Θ_{u})$ .
$Θ_{u} \subset X$ is a section of the semiflow and the first return map to $Θ_{u}$ is an expanding irreducible train track map representing $φ_{u} \in Out (ker (u))$ such that $G = ker (u) ⋊_{φ_{u}} ℤ$ .
The logarithm of the stretch factor of $φ_{u}$ is precisely $ℌ (u)$ .
If $φ$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u \in A$ the automorphism $φ_{u}$ of $ker (u)$ is also hyperbolic and fully irreducible.

Keywords

train track map, free-by-cyclic group, entropy

Mathematical Subject Classification 2010

Primary: 20F65

References

Publication

Received: 6 June 2014

Revised: 30 December 2014

Accepted: 26 January 2015

Published: 20 October 2015

Proposed: Walter Neumann

Seconded: Benson Farb, Danny Calegari

Authors

Spencer Dowdall
Department of Mathematics Vanderbilt University 1326 Stevenson Center Nashville, TN 37240 USA
http://www.math.vanderbilt.edu/~dowdalsd/
Ilya Kapovich
Department of Mathematics University of Illinois at Urbana-Champaign 1409 West Green Street Urbana, IL 61801 USA
http://www.math.uiuc.edu/~kapovich/
Christopher J Leininger
Department of Mathematics University of Illinois at Urbana-Champaign 1409 West Green Street Urbana, IL 61801 USA
http://www.math.uiuc.edu/~clein/

Volume 19, issue 5 (2015)