Axes in Outer Space

Axes in Outer Space 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 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.