## Abstract

Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space CV (F _{k} ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k - 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs.

Original language | English (US) |
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Pages (from-to) | 1201-1236 |

Number of pages | 36 |

Journal | Geometric and Functional Analysis |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2007 |

## Keywords

- Free groups
- Geodesic currents
- Metric graphs
- Patterson-Sullivan measures
- Volume entropy

## ASJC Scopus subject areas

- Analysis
- Geometry and Topology