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 ≅ ℒ_ℚ(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 ℒ_ℚ(G) is not isomorphic to L _ℚ(G) as Lie algebras.

Key Words:

• Baker–Campbell–Hausdorff formula
• Mal'cev completion
• Quasi-isometry
• Relatively free groups
• Relatively free Lie algebras
• Varieties of groups
• Varieties of Lie algebras

2000 Mathematics Subject Classification:

• Primary 17B30
• Secondary 17B01, 17B40

Notes

Communicated by I. Shestakov.