Algebraic Structure of Tensor Completions of 2-Nilpotent Torsion-Free Groups

We study tensor completions G⊗N2,RR of finitely generated torsion-free 2-nilpotent groups G in the class N2,R of all 2-nilpotent R-groups over a binomial domain R. We show that G⊗N2,RR is isomorphic to the group (G⊗HR) × D, where G⊗HR is the classical Hall R-completion of G, D is an Abelian R-group, and the direct product is a product of abstract groups (not R-groups!). In particular, this answers an old question of Remeslennikov about the algebraic structure of free 2-nilpotent R-groups in the quasivariety N2,R (they are precisely the tensor R-completions of free 2-nilpotent groups).