On the theories of free solvable groups Original Research Article

We prove that the terms of the derived series of a free solvable group are definable by existential formulae. We use this result to prove some ‘model theoretic’ results about free solvable groups. For example, we prove that if Hilbertʹs 10th problem has a negative answer for the field of the rationals, then the universal theory of a noncyclic free solvable group of class ≥ 3 is undecidable.