### Abstract

Let X be a geodesic metric space. Gromov proved that there exists ε_{0} > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε_{0} · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε_{0}. We also show that if all the triangles Δ ⊆ X satisfy the Rips condition with constant ε_{0} · pr(Δ), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.

Original language | English |
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Pages (from-to) | 139-149 |

Number of pages | 11 |

Journal | Geometriae Dedicata |

Volume | 142 |

Issue number | 1 |

Early online date | 12 Mar 2009 |

DOIs | |

Publication status | Published - Oct 2009 |

### Keywords

- Asymptotic cone
- Detour
- Gromov-hyperbolic
- Real tree
- Rips condition

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Geometriae Dedicata*,

*142*(1), 139-149. https://doi.org/10.1007/s10711-009-9363-4