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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)
DOI: https://doi.org/10.1090/S0065-9266-2011-00620-9
Published electronically: March 10, 2011
MSC: Primary 20F65, 57M07
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
Abstract
We 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 our 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. We 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 we prove their equivalence.- Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. MR 1465330, DOI 10.1007/s002220050168
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