## Abstract

For the free group F_{N} of finite rank N ≥ 2 we construct a canonical Bonahon-type, continuous and Out(F_{N})-invariant geometric intersection form Here cv(F_{N}) is the closure of unprojectivized Culler-Vogtmann Outer space cv (F_{N}) in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv(F_{N}) consists of all very small minimal isometric actions of F_{N} on R-trees. The projectivization of cv(F_{N}) provides a free group analogue of Thurston's compactification of Teichmüller space. As an application, using the intersection graph determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.

Original language | English (US) |
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Pages (from-to) | 1805-1833 |

Number of pages | 29 |

Journal | Geometry and Topology |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - 2009 |

## Keywords

- Curve complex
- Free group
- Geodesic current
- Outer space

## ASJC Scopus subject areas

- Geometry and Topology