## Abstract

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion-free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first-order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion-free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent.

Original language | American English |
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Pages (from-to) | 217-273 |

Number of pages | 57 |

Journal | Proceedings of the London Mathematical Society |

Volume | 99 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2009 |

### Bibliographical note

Funding Information:The work that is presented in this paper was partially supported by an Israel academy of sciences fellowship, and NSF grant no. DMS9729992 through the IAS.