# real tree

A metric space $X$ is said to be a *real tree* or
*$\mathrm{R}$-tree*, if for each $x,y\in X$ there is a unique arc
from $x$ to $y$, and furthermore this arc is an isometric http://planetmath.org/node/429embedding^{}.

Every real tree is a hyperbolic metric space; moreover, every real tree is 0 hyperbolic.

The Cayley graph^{} of any free group^{} is considered to be a real tree. Note that its graph is a tree in the graph theoretic sense. To make it a real tree, we view the edges as isometric (http://planetmath.org/Isometric) to the line segment $[0,1]$ under a (surjective^{}) isometry (http://planetmath.org/Isometry) and attach the edges to the tree. The resulting 1-complex is then a locally finite^{} real tree. Because of this result, every free group is a hyperbolic group.

Title | real tree |

Canonical name | RealTree |

Date of creation | 2013-03-22 15:16:55 |

Last modified on | 2013-03-22 15:16:55 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 10 |

Author | GrafZahl (9234) |

Entry type | Definition |

Classification | msc 54E99 |

Classification | msc 54E40 |

Synonym | $\mathbb{R}$-tree |

Related topic | MetricSpace |

Related topic | Arc |

Related topic | Curve |

Related topic | SNCFMetric |

Related topic | Isometry |

Related topic | FreeGroup |

Related topic | HyperbolicGroup |