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For a torsion free finitely generated nilpotent group G we naturallyassociate four finite dimensional nilpotent Lie algebras over a field ofcharacteristic zero. We show that if G is a relatively free group of somevariery of nilpotent groups then all the above Lie algebras are isomorphic. Asa result, any two quasi-isometric relatively free nilpotent groups areisomorphic.  Moreover let L be a relatively free nilpotent Lie algebra over Q generated byX. We give L the structure of a group by means of the Baker-Campbell-Hausdorffformula and we show that the subgroup H generated by X is relatively free insome variety of nilpotent groups, is Magnus and certain Lie algebras associatedto H are isomorphic. This isomorphism is extended to relatively free residuallytorsion-free nilpotent groups. Finally, we give an example that demonstratesthat this is not always the case with finitely generated Magnus nilpotentgroups.

Relatively free nilpotent torsion-free groups and their Lie algebras