Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

Z. Sela*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

57 Scopus citations


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 languageAmerican English
Pages (from-to)217-273
Number of pages57
JournalProceedings of the London Mathematical Society
Issue number1
StatePublished - 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.


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