Abstract
We develop a notion of axis in the Culler-Vogtmann outer space X r of a finite rank free group F r, with respect to the action of a nongeometric, fully irreducible outer automorphism φ. Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmüller space, X r has no natural metric, and φ seems not to have a single natural axis. Instead our axes for φ, while not unique, fit into an "axis bundle" A φ with nice topological properties: A φ is a closed subset of X r proper homotopy equivalent to a line, it is invariant under φ, the two ends of A φ limit on the repeller and attractor of the source-sink action of φ on compactified outer space, and A φ depends naturally on the repeller and attractor. We propose various definitions for A φ, each motivated in different ways by train track theory or by properties of axes in Teichmüller space, and we prove their equivalence.
Original language | English (US) |
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Pages (from-to) | 1-110 |
Number of pages | 110 |
Journal | Memoirs of the American Mathematical Society |
Volume | 213 |
Issue number | 1004 |
DOIs | |
State | Published - Sep 2011 |
All Science Journal Classification (ASJC) codes
- General Mathematics
- Applied Mathematics