Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Abstract

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ℒ K (G), grad(ℓ)(ℒ K (G)), grad(g)(exp ℒ K (G)), and L K (G). Let 𝔗 c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F n (𝔗 c ) be the relatively free group of rank n in 𝔗 c . We prove that ℒ K (F n (𝔗 c )) is relatively free in some variety of nilpotent Lie algebras, and ℒ K (F n (𝔗 c )) ≅ L K (F n (𝔗 c )) ≅ grad(ℓ)(ℒ K (F n (𝔗 c ))) ≅ grad(g)(exp ℒ K (F n (𝔗 c ))) as Lie algebras in a natural way. Furthermore, F n (𝔗 c ) is a Magnus nilpotent group. Let G 1 and G 2 be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G 1 and G 2 are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ℚ of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ≅ ℒℚ(H) ≅ L ℚ(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ℒ K and L K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ℒℚ(G) is not isomorphic to L ℚ(G) as Lie algebras.