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Axes in Outer Space
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Axes in Outer Space (Paperback)

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Synopsis

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 Teichmuller 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 Teichmuller space, and they prove their equivalence.

Science & MathematicsMathematicsGeometry Publisher: American Mathematical Society Publication Date: 15/12/2012 ISBN-13: 9780821869277  Details: Type: Paperback Format: Books
Availability: To Order. Estimated despatch in 1-3 weeks.  

Michael Handel is at CUNY, Herbert H. Lehman College, Bronx, NY

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