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

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.

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