Axes in Outer Space

ISBN-13: 9780821869277 / Angielski / Miękka / 2012 / 104 str.

Michael Handel

Opis

The authors develop a notion of axis in the Culler-Vogtmann outer space $mathcal_r$ 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, $mathcal_r$ 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"" $mathcal_phi$ with nice topological properties: $mathcal_phi$ is a closed subset of $mathcal_r$ proper homotopy equivalent to a line, it is invariant under $phi$, the two ends of $mathcal_phi$ limit on the repeller and attractor of the source-sink action of $phi$ on compactified outer space, and $mathcal_phi$ depends naturally on the repeller and attractor.

The authors propose various definitions for $mathcal_phi$, each motivated in different ways by train track theory or by properties of axes in Teichmuller space, and they prove their equivalence.