# hyperbolic group

A finitely generated group $G$ is *hyperbolic* if, for some finite set of generators^{} $A$ of $G$, the Cayley graph^{} $\mathrm{\Gamma}(G,A)$, considered as a metric space with $d(x,y)$ being the minimum number of edges one must traverse to get from $x$ to $y$, is a hyperbolic metric space.

Hyperbolicity is a group-theoretic property. That is, if $A$ and $B$ are finite sets of generators of a group $G$ and $\mathrm{\Gamma}(G,A)$ is a hyperbolic metric space, then $\mathrm{\Gamma}(G,B)$ is a hyperbolic metric space.

examples of hyperbolic groups include finite groups^{} and free groups^{}. If $G$ is a finite group, then for any $x,y\in G$, we have that $d(x,y)\le |G|$. (See the entry Cayley graph of ${S}_{3}$ (http://planetmath.org/CayleyGraphOfS_3) for a pictorial example.) If $G$ is a free group, then its Cayley graph is a real tree.

Title | hyperbolic group |
---|---|

Canonical name | HyperbolicGroup |

Date of creation | 2013-03-22 17:11:43 |

Last modified on | 2013-03-22 17:11:43 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 6 |

Author | Wkbj79 (1863) |

Entry type | Definition |

Classification | msc 05C25 |

Classification | msc 20F06 |

Classification | msc 54E35 |

Synonym | hyperbolicity |

Related topic | RealTree |