Volume 19, issue 5 (2015)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 26
Issue 2, 477–936
Issue 1, 1–476

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Dynamics on free-by-cyclic groups

Spencer Dowdall, Ilya Kapovich and Christopher J Leininger

Geometry & Topology 19 (2015) 2801–2899

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.
train track map, free-by-cyclic group, entropy
Mathematical Subject Classification 2010
Primary: 20F65
Received: 6 June 2014
Revised: 30 December 2014
Accepted: 26 January 2015
Published: 20 October 2015
Proposed: Walter Neumann
Seconded: Benson Farb, Danny Calegari
Spencer Dowdall
Department of Mathematics
Vanderbilt University
1326 Stevenson Center
Nashville, TN 37240
Ilya Kapovich
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 West Green Street
Urbana, IL 61801
Christopher J Leininger
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 West Green Street
Urbana, IL 61801