Geometric equivalence of $\pi$-torsion free nilpotent groups

In this paper we study the property of geometric equivalence of groups introduced by B. Plotkin \cite{P1, P2}. Sufficient and necessary conditions are presented for a $\pi$-torsion-free nilpotent group to be geometrically equivalent to its $\pi$-completion. We prove that a relatively free nilpotent $\pi$-torsion-free group and its $\pi$-completion define the same quasi-variety. Examples of $\pi$-torsion-free nilpotent groups that are geometrically equivalent to their $\pi$-completions are given.