On the theories of free solvable groups
In 1960, Malcev proved that a noncyclic free solvable group \(S\) of class \(\geq 2\) has an undecidable theory. The author proves that if Hilbert's 10th problem has a negative answer for the field of rationals (still a famous open question), then the universal theory of \(S\) in the case of class \(\geq 3\) is undecidable. Remind that the universal theory of a free metabelian group is decidable by the author's known theorem. Note that by the reviewer's result Hilbert's 10th problem for the rationals is equivalent to the problem of the decidability of the universal theory of a noncyclic free nilpotent group of class \(\geq 2\). To derive his main theorems the author proves that the terms of the derived series of a free solvable group are definable by existential formulae. Malcev defined them by universal ones.
scientific article; zbMATH DE number 1308222
scientific article