Elementary Equivalence and Profinite Completions: a Characterization of Finitely Generated Abelian-by-finite Groups

In this paper, we show that any finitely generated abelian-byfinite group is an elementary submodel of its profinite completion. It follows that two finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images. We give an example of two finitely generated abelian-by-finite groups G, H which satisfy these properties while G x Z and H x Z are not isomorphic. We also prove that a finitely generated nilpotent-by-finite group is elementarily equivalent to its profinite completion if and only if it is abelian-by-finite. The definitions and results of model theory which are used here, and in particular the notions of elementary equivalence and elementary submodel, are given in [2]. Anyhow, for the sake of brevity, we denote by Gu the ultrapower G1 /U, for each group G and each ultrafilter U over a set /. Concerning group theory, the reader is referred to [9 and 10]. We obtain the following characterization: THEOREM. A finitely generated nilpotent-by-finite group is elementarily equivalent to its profinite completion if and only if it is abelian-by-finite. This result is a consequence of Theorems 1 and 2 below. THEOREM l. Any finitely generated abelian-by-finite group is an elementary submodel of its profinite completion. THEOREM 2. If G is a finitely generated nilpotent-by-finite, but not abelian-byfinite, group, then there exists an existential sentence, built up from the multiplication symbol, which is false in G and true in the profinite completion of G. Theorem 1 generalizes [7, Theorem 1.4], which was only valid for finitely generated finite-by-abelian groups. A finitely generated finite-by-abelian group is abelian-by-finite, since its center is a normal subgroup of finite index according to the result of P. Hall which is mentioned in [10, p. 12]. On the other hand, the finitely presented abelian-by-finite group (x,y; y2 = l,y_1xy = x_1) is not finite-by-abelian as its center is trivial. By [11, Theorem 2.1 and 6, Theorem 5.5], the following properties are equivalent for two finitely generated finite-by-abelian groups G, H: (1) G and H are elementarily equivalent; Received by the editors July 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 20A15; Secondary 03C60, 20F18. ©1988 American Mathematical Society 0002-9939/88 $1.00 -I$ 25 per page