International Congress of Mathematicians

Diophantine Geometry over Groups and the Elementary Theory • • • 91Theorem 10 ([Se8],7.2). Let Li,L 2 be uniform lattices in real semi-simple Liegroups that are not SL 2 (R). Then Li is elementary equivalent to F 2 if and only ifLi and F 2 are conjugate lattices in the same real Lie group G.Proposition 9 shows that rigid hyperbolic groups are elementary equivalentif and only if they are isomorphic. To classify elementary equivalence classes ofhyperbolic groups in general, we associate with every (torsion-free) hyperbolic groupF, a subgroup of it, that we call the elementary core of F, and denote EC(T). Theelementary core is a retract of the ambient hyperbolic group F, and although it isnot canonical, its isomorphism type is an invariant of the ambient hyperbolic group.The elementary core is constructed iteratively from the ambient hyperbolic groupas we describe in definition 7.5 in [Se8].The elementary core of a hyperbolic group is a prototype for its elementary theory.Theorem 11 ([Se8],7.6). LetT be a non-elementary torsion-free hyperbolic groupthat is not a oj-residually free tower, i.e., that is not elementary equivalent to a freegroup. Then T is elementary equivalent to its elementary core EC(T). Furthermore,the embedding of the elementary core EC(T) in the ambient group T is anelementary embedding.Finally, the elementary core is a complete invariant of the class of groups thatare elementary equivalent to a given (torsion-free) hyperbolic group.Theorem 12 ([Se8],7.9). Le£Fi,F 2 be two non-elementary torsion-free hyperbolicgroups. Then Fi and F 2 are elementary equivalent if and only if their elementarycores EC(T'i) and FC(F 2 ) are isomorphic.Theorem 12 asserts that the elementary class of a torsion-free hyperbolic groupis determined by the isomorphism class of its elementary core. Hence, in order to beable to decide whether two torsion-free hyperbolic groups are elementary equivalentone needs to compute their elementary core, and to decide if the two elementarycoresare isomorphic. Both can be done using the solution to the isomorphismproblem for torsion-free hyperbolic groups.Theorem 13 ([Se8],7.11). LetT'i,T2 be two torsion-free hyperbolic groups. Thenit is decidable if T\ is elementary equivalent to F 2 .References[Hr] E. Hrushovski, private communication.[Kh-My] O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over afree group II, Jour, of Algebra 200 (1998), 517^570.[Mai] G. S. Makanin, Equations in a free group, Math. USSR Izvestiya 21(1983), 449-169.[Ma2], Decidability of the universal and positive theories of a free group,Math. USSR Izvestiya 25 (1985), 75^88.