Mekler's construction preserves \(CM\)-triviality
In a famous paper, \textit{A. Mekler} [J. Symb. Log. 46, 781-788 (1981; Zbl 0482.03014)] showed that 2-nilpotent groups of prime exponent \(p>2\) interpret a class of graphs (called nice) and, through them, all the structures of an arbitrary finite language \(L\). Mekler's construction preserves elementary equivalence (and so produces, for every complete theory \(S\) of \(L\)-structures, a complete theory \(T_S\) of groups as before interpreting \(S\) in a way independent of the particular choice of \(S\)), as well as the stability spectrum (hence \(T_S\) is stable, superstable, \(\omega\)-stable if and only if \(S\) is). Transfer properties of other relevant notions -- such as presentability NDOP, depth, NOTOP -- were discussed in this setting by A. Baudisch in some previous papers. Here it is shown that Mekler's construction preserves also simplicity (for arbitrary structures) and \(CM\)-triviality (for stable structures), so two basic notions in modern model theory, both born or rediscovered some years after Mekler's original paper as an enlargement of stability and one-basedness respectively. The part concerning simplicity is a common work of the author with his student Alexander Pentzel. Clearly these results can also be used to produce examples of stable \(CM\)-trivial groups; in particular, it is shown that the free group \(F_2 (p, \omega)\) in the variety of 2-nilpotent groups of exponent \(p>2\) with \(\omega\) free generators is \(CM\)-trivial. The paper is written is a very accurate way, and includes an exhaustive and clear preliminary outline of the main facts concerning Mekler's construction and \(CM\)-triviality. The author goes into Mekler's treatment and develops a finer analysis, handling the new involved notions.
scientific article; zbMATH DE number 1787754
scientific article