RELATIVELY FREE NILPOTENT TORSION-FREE GROUPS AND THEIR LIE ALGEBRAS

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 (sic)(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(sic)(c) be the relatively free group of rank n in (sic)(c). We prove that L-K(F-n((sic)(c))) is relatively free in some variety of nilpotent Lie algebras, and L-K(F-n((sic)(c))) congruent to L-K(F-n((sic)(c) congruent to grad((l)) (L-K(F-n((sic)(c))) congruent to grad((g)) (exp L-K(F-n((sic)(c))))as Lie algebras in a natural way. Furthermore, F-n((sic)(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 congruent to L-Q(H) congruent to 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.