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Axes in Outer Space
Michael Handel, CUNY, Herbert H. Lehman College, Bronx, NY, and Lee Mosher, Rutgers University, Newark, NJ
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Memoirs of the American Mathematical Society
2011; 104 pp; softcover
Volume: 213
ISBN-10: 0-8218-6927-2
ISBN-13: 978-0-8218-6927-7
List Price: US$70 Individual Members: US$42
Institutional Members: US\$56
Order Code: MEMO/213/1004

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.