Profinite Completion of Free Pro-$\mathcal{C}$ Groups

TL;DR

This paper studies the properties of free pro-$ ext{ extbf{C}}$ groups under profinite completion, showing that for certain varieties, the completion preserves the free pro-$ ext{ extbf{C}}$ structure and rank.

Contribution

It proves that for some varieties, the profinite completion of a free pro-$ ext{ extbf{C}}$ group of infinite rank remains free of the same rank.

Findings

Profinite completion preserves freeness for certain varieties.

The rank of free pro-$ ext{ extbf{C}}$ groups is maintained after completion.

The paper characterizes varieties where this property holds.

Abstract

We investigate the ability of a free pro-$\CC$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\CC$, the profinite completion of any order, of a free pro-$\CC$ group of infinite rank, is a free pro-$\CC$ group as well, of the corresponding rank.