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\(\exists\)-free groups and groups with length function. (English) Zbl 0856.20002
Bokut’, L. A. (ed.) et al., Second international conference on algebra dedicated to the memory of A. I. Shirshov. Proceedings of the conference on algebra, August 20-25, 1991, Barnaul, Russia. Providence, RI: American Mathematical Society. Contemp. Math. 184, 369-376 (1995).
Let \(F_n\) be a free non-abelian group of rank \(n\). Since \(\text{Th}_\exists F_{n_1}=\text{Th}_\exists F_{n_2}\) for any pair \((n_1,n_2)\) one can speak of the \(\exists\)-theory of a non-abelian free group \(\text{Th}_\exists F\). A group \(G\) is said to be \(\exists\)-free if \(\text{Th}_\exists G=\text{Th}_\exists F\).
Let \((n,k)\) be a pair of natural numbers such that \(n\geq k\). By \(\Lambda_{nk}\) is denoted an ordered abelian group with the following properties: the group \(\Lambda_{nk}\) is the direct sum of \(n\) copies of \(Z\), and the total series of convex subgroups of \(\Lambda_{nk}\) contains \(k\) non-trivial terms. It is known [V. N. Remeslennikov, Ukr. Mat. Zh. 44, No. 6, 813-818 (1992; Zbl 0784.20015), and independently A. M. Gaglione and D. Spellman, Contemp. Math. 169, 277-281 (1994; see the following review Zbl 0856.20003)] that for every finitely generated \(\exists\)-free group \(G\) there exists a pair \((n,k)\) of natural numbers, and an abelian ordered group \(\Lambda_{nk}\) such that there exists a \(\Lambda_{nk}\)-valued Lyndon length function making \(G\) into a \(\Lambda_{nk}\)-free group in the sense of H. Bass.
A notion of \(\exists\)-free constructible group is introduced, and an example of an \(\exists\)-free constructible non-\(\exists\)-free group is presented. This group is \(\Lambda_{22}\)-free and hence is a counterexample to the question posed by A. M. Gaglione and D. Spellman in the paper mentioned above on the equivalence of the notions of \(\exists\)-free groups and \(\Lambda_{nk}\)-free groups for the class of all finitely generated groups.
For the entire collection see [Zbl 0824.00029].

20A15 Applications of logic to group theory
20E05 Free nonabelian groups
20F05 Generators, relations, and presentations of groups
03C60 Model-theoretic algebra