Electronic ISBN:  9781470406219 
Product Code:  MEMO/213/1004.E 
104 pp 
List Price:  $74.00 
MAA Member Price:  $66.60 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 213; 2011MSC: 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 pseudoAnosov 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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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 pseudoAnosov 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