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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 (2000): 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 of a finite rank free group , with respect to the action of a nongeometric, fully irreducible outer automorphism . Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmüller space, has no natural metric, and seems not to have a single natural axis. Instead our axes for , while not unique, fit into an “axis bundle” with nice topological properties: is a closed subset of proper homotopy equivalent to a line, it is invariant under , the two ends of limit on the repeller and attractor of the source–sink action of on compactified outer space, and depends naturally on the repeller and attractor. We propose various definitions for , 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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