## Abstract

We prove an ergodic rigidity theorem for discrete isometry groups of CAT(-1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2-polyhedra, hyperbolic Bruhat-Tits buildings and rank one symmetric spaces. We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least 2π, on a closed surface, with boundary map absolutely continuous with respect to the Patterson-Sullivan measures, are isometric. For that, we generalize J.-P. Otal's result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least 2π, on a closed surface, is determined, up to isometry, by its marked length spectrum.

Original language | English |
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Pages (from-to) | 349-388 |

Number of pages | 40 |

Journal | Commentarii Mathematici Helvetici |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 1997 |

Externally published | Yes |

## Keywords

- Discrete group
- Divergence group
- Hyperbolic building
- Marked length spectrum
- Negative curvature
- Patterson-sullivan measure
- Rigidity
- Surface with conical singularities

## ASJC Scopus subject areas

- Mathematics (all)