## Abstract

We study those fully irreducible outer automorphisms f of a finite rank free group F_{r} which are parageometric, meaning that the attracting fixed point of φ in the boundary of outer space is a geometric R-tree with respect to the action of F_{r}, but φ itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface. Our main result shows that the expansion factor of φ is strictly larger than the expansion factor of φ^{-1}. As corollaries (proved independently by Guirardel), the inverse of a parageometric outer automorphism is neither geometric nor parageometric, and a fully irreducible outer automorphism φ is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric R-trees.

Original language | English (US) |
---|---|

Pages (from-to) | 3153-3183 |

Number of pages | 31 |

Journal | Transactions of the American Mathematical Society |

Volume | 359 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2007 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics