Relatively free nilpotent torsion-free groups and their Lie algebras

TL;DR

This paper explores the relationship between relatively free nilpotent groups and their associated Lie algebras, establishing isomorphisms and properties that connect group and Lie algebra structures in the context of nilpotent varieties.

Contribution

It proves that all associated Lie algebras are isomorphic for relatively free nilpotent groups and extends these results to residually torsion-free nilpotent groups, highlighting key structural insights.

Findings

All associated Lie algebras are isomorphic for relatively free nilpotent groups.

Relatively free nilpotent groups are quasi-isometric if and only if they are isomorphic.

An example shows not all finitely generated Magnus nilpotent groups share this property.

Abstract

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relatively free in some variety of nilpotent groups, is Magnus and certain Lie algebras associated to H are isomorphic. This isomorphism is extended to relatively free residually torsion-free nilpotent groups. Finally, we give an example that demonstrates that this is not always the case with finitely generated Magnus nilpotent groups.