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Diophantine geometry over groups VII: The elementary theory of a hyperbolic group
Author(s) -
Sela Z.
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn052
Subject(s) - mathematics , free group , torsion (gastropod) , hyperbolic group , diophantine equation , relatively hyperbolic group , elementary theory , invariant (physics) , pure mathematics , group (periodic table) , algebra over a field , discrete mathematics , hyperbolic manifold , hyperbolic function , mathematical analysis , medicine , chemistry , surgery , organic chemistry , computer science , programming language , mathematical physics
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.