The Relatively Free Groups F(Nc∧A2) Satisfy Noncentral Commutative Transitivity

We prove that a free group, F(Nc∧A2), relative to the variety, Nc∧A2, of all groups simultaneously nilpotent of class at most c and metabelian is such that the centralizer of every noncentral element is abelian. We relate that result to the model theory of such groups as well as a quest to find a relative analog in Nc∧A2 of a classical theorem of Benjamin Baumslag. We also touch briefly on similar considerations in the varieties Nc of nilpotent groups