## Abstract

We prove that for k > 5 there does not exist a continuous map ∂CV(F_{k}) → PCurr(F_{k}) that is either Out(F _{k})-equivariant or Out(F_{k})-anti-equivariant. Here ∂CV(F_{k}) is the 'length function' boundary of Culler-Vogtmann's Outer space CV(F_{k}), and PCurr(F_{k}) is the space of projectivized geodesic currents for F_{k}. We also prove that, if k ≥ 3, for the action of Out(F_{k}) on ℙCwrr(F_{k}) and for the diagonal action of Out(F_{k}) on the product space ∂CV(F _{k}) × PCurr(F_{k}), there exist unique non-empty minimal closed Out(F_{k})-invariant sets. Our results imply that for k ≥ 3 any continuous Out(F_{k})-equivariant embedding of CV(F_{k}) into ℙCurr(F_{k}) (such as the Patterson-Sullivan embedding) produces a new compactification of Outer space, different from the usual 'length function' compactification CV(F_{k}) = CV(F_{k}) ∪ ∂CV(F_{k}).

Original language | English (US) |
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Pages (from-to) | 827-847 |

Number of pages | 21 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2007 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics