# Axes in Outer Space

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*Michael Handel; Lee Mosher*

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.