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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
List Price: $74.00 MAA Member Price:$66.60
AMS Member Price: $44.40 Click above image for expanded view 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  List Price:$74.00 MAA Member Price: $66.60 AMS Member Price:$44.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 2132011; 104 pp
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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 2132011; 104 pp
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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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