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Axes in Outer Space
 
Michael Handel CUNY, Herbert H. Lehman College, Bronx, NY
Lee Mosher Rutgers University, Newark, NJ
Front Cover for Axes in Outer Space
Available Formats:
Electronic ISBN: 978-1-4704-0621-9
Product Code: MEMO/213/1004.E
104 pp 
List Price: $74.00
MAA Member Price: $66.60
AMS Member Price: $44.40
Front Cover for Axes in Outer Space
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Axes in Outer Space
Michael Handel CUNY, Herbert H. Lehman College, Bronx, NY
Lee Mosher Rutgers University, Newark, NJ
Available Formats:
Electronic ISBN:  978-1-4704-0621-9
Product Code:  MEMO/213/1004.E
104 pp 
List Price: $74.00
MAA Member Price: $66.60
AMS Member Price: $44.40
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2132011
    MSC: Primary 20; Secondary 57;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Preliminaries
    • 3. The ideal Whitehead graph
    • 4. Cutting and pasting local stable Whitehead graphs
    • 5. Weak train tracks
    • 6. Topology of the axis bundle
    • 7. Fold lines
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Volume: 2132011
MSC: Primary 20; Secondary 57;

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.

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