Axes in outer space

✍ Scribed by Michael Handel, Lee Mosher

• Publisher: Amer Mathematical Society
• Year: 2011
• Tongue: English
• Leaves: 117
• Series: Memoirs of the American Mathematical Society 1004
• Category: Library

✦ 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.