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 Teichmueller 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 "e;axis bundle"e; $mathcal{A}_phi$ 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 Teichmueller space, and they prove their equivalence.
| ISBN: | 9781470406219 |
| Publication date: | 30th November -0001 |
| Author: | Handel, Michael |
| Publisher: | American Mathematical Society |
| Format: | Ebook |
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 Teichmueller 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 "e;axis bundle"e; $mathcal{A}_phi$ 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 Teichmueller space, and they prove their equivalence.
Axes in Outer Space is available in Ebook
Axes in Outer Space was written by Handel, Michael and published by American Mathematical Society
£88.80