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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

algerba/algebra

The Relatively Free Groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>∧</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> Satisfy Noncentral Commutative Transitivity

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