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Proceedings of the London Mathematical Society Advance Access originally published online on February 23, 2009
Proceedings of the London Mathematical Society 2009 99(1):217-273; doi:10.1112/plms/pdn052
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© 2009 London Mathematical Society

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

Z. Sela

Mathematics Department
Hebrew University
Jerusalem 91904
Israel

Received 27 September 2006. Revision received 9 September 2008.

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.

2000 Mathematics Subject Classification 20F65 (03B35,20E05,20F10).

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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