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Axes in outer space

About this Title

Michael Handel, Lehman College, CUNY and Lee Mosher, Rutgers University, Newark

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 213, Number 1004
ISBNs: 978-0-8218-6927-7 (print); 978-1-4704-0621-9 (online)
Published electronically: March 10, 2011
MathSciNet review: 2858636
MSC: Primary 20F65, 57M07

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. The ideal Whitehead graph
  • Chapter 4. Cutting and pasting local stable Whitehead graphs
  • Chapter 5. Weak train tracks
  • Chapter 6. Topology of the axis bundle
  • Chapter 7. Fold lines


We develop a notion of axis in the Culler-Vogtmann outer space $\mathcal{X}_r$ of a finite rank free group $F_r$, with respect to the action of a nongeometric, fully irreducible outer automorphism $\phi$. Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmüller space, $\mathcal{X}_r$ has no natural metric, and $\phi$ seems not to have a single natural axis. Instead our axes for $\phi$, while not unique, fit into an ``axis bundle'' $\mathcal{A}_\phi$ with nice topological properties: $\mathcal{A}_\phi$ is a closed subset of $\mathcal{X}_r$ proper homotopy equivalent to a line, it is invariant under $\phi$, the two ends of $\mathcal{A}_\phi$ limit on the repeller and attractor of the source-sink action of $\phi$ on compactified outer space, and $\mathcal{A}_\phi$ depends naturally on the repeller and attractor. We propose various definitions for $\mathcal{A}_\phi$, each motivated in different ways by train track theory or by properties of axes in Teichmüller space, and we prove their equivalence.

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