Axiomatization and other model-theoretic properties of groups

[22] An application of Ramsey's theorem to groups with homogeneous theory, Communications in Algebra 28 (2000), 2977-2981.
[23] Axiomatization of abelian-by-G groups for a finite group G, Archive for Mathematical Logic 40 (2001), 515-521.
[24] Elementary equivalence for abelian-by-finite and nilpotent groups, Journal of Symbolic Logic 66 (2001), 1471-1480.
[26] Quasi-finitely axiomatizable nilpotent groups (with G. Sabbagh), Journal of Group Theory 9 (2006), 95-106.
[27] Quasi-finitely axiomatizable groups and groups which are prime models, Journal of Group Theory 9 (2006), 107-116.

[30] Some new examples of quasi-finitely axiomatizable groups which are prime models, 5 pages, in preparation.

    In [22], for any integers n>m≥2, we say that a complete theory T is (m,n)-homogeneous if, for each model M of T, any n-tuples a,b in M have the same type if the corresponding m-tuples from a,b have the same type. It was conjectured by H. Kikyo that, if M is an infinite group, with possibly additional structure, then the theory of M is not (m,n)-homogeneous. We prove a general result on structures with (m,n)-homogeneous theory which implies that, if M is a counterexample to this conjecture, then there exists an integer h such that each abelian subgroup of M has at most h elements. It follows that there exist an integer k such that M^k=1, and an integer l such that each finite subgroup of M has at most l elements.
    In [23], we show that, for each finite group G, there exists an axiomatization of the class of abelian-by-G groups with a unique sentence. This result had been conjectured by A. Marcja and C. Toffalori. In the proof, we use the definability of the subgroups M^n in an abelian-by-finite group M, and the Auslander-Reiten sequences for modules over an Artin algebra.
    In [24], we show that two abelian-by-finite groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. We also prove that abelian-by-finite groups satisfy a quantifier elimination property. On the other hand, for each integer n, we give some examples of nilpotent groups which satisfy the same sentences with n alternations of quantifiers and do not satisfy the same sentences with n+1 alternations of quantifiers.

    More recently, new techniques introduced by A. Morozov et A. Nies made it possible to have a better insight into the model-theoretic properties of finitely generated groups. According to their definition, we say that a finitely generated group G is quasi-finitely axiomatizable, or more briefly QFA, if it satisfies a sentence φ such that each finitely generated group H which satisfies φ is isomorphic to G. We say that G is a prime model of its theory, or more briefly that G is prime, if it is isomorphic to an elementary submodel of any model of its theory.
    In [26] and [27], we investigate the relations between these two properties, and their algebraic consequences. For finitely generated nilpotent-by-finite groups, we prove that the two properties are equivalent and we characterize them algebraically. It follows that finitely generated free nilpotent groups and groups of upper unitriangular nxn matrices for n≥3 are QFA, which gives a positive answer to a question of A. Nies, and that they are prime, which disproves a conjecture of O.V. Belegradek.

    In [30], we consider groups which are semi-direct products of a finitely generated torsion-free abelian group by a cyclic group. We investigate the following questions: Are they quasi-finitely axiomatizable (QFA)? Are they prime models of their theory? Are they elementarily equivalent?
    It follows from the results of A. Morozov et A. Nies that there exist some finitely generated prime groups which are not QFA. On the other hand, we do not know presently if a QFA group is necessarily prime. Also one can wonder whether there exists an infinite QFA group, or more generally an infinite finitely generated group, with a finitely axiomatizable theory.