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

TL;DR

This paper investigates when $\pi$-torsion-free nilpotent groups are geometrically equivalent to their $\pi$-completions, providing conditions, examples, and showing they define the same quasi-variety.

Contribution

It establishes necessary and sufficient conditions for geometric equivalence between $\pi$-torsion-free nilpotent groups and their $\pi$-completions, including the case of relatively free groups.

Findings

Conditions for geometric equivalence are characterized.

Relatively free nilpotent $\pi$-torsion-free groups and their $\pi$-completions define the same quasi-variety.

Examples of such groups are provided.

Abstract

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.