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Axes in Outer Space
Michael Handel, CUNY, Herbert H. Lehman College, Bronx, NY, and Lee Mosher, Rutgers University, Newark, NJ
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Memoirs of the American Mathematical Society
2011; 104 pp; softcover
Volume: 213
ISBN-10: 0-8218-6927-2
ISBN-13: 978-0-8218-6927-7
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/213/1004
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The authors 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 these 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.

The authors 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 they prove their equivalence.

Table of Contents

  • Introduction
  • Preliminaries
  • The ideal Whitehead graph
  • Cutting and pasting local stable Whitehead graphs
  • Weak train tracks
  • Topology of the axis bundle
  • Fold lines
  • Bibliography
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