Axes in Outer Space

(Memoirs of the American Mathematical Society)

5bdeaa4c3ec0f.jpg Author Michael Handel
Isbn 9780821869277
File size 1.04MB
Year 2011
Pages 104
Language English
File format PDF
Category Mathematics

Book Description:

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.



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