  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.
