We investigate the ability of a free pro-[Formula: see text] group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties [Formula: see text], the profinite completion of any order, of a free pro-[Formula: see text] group of infinite rank, is a free pro-[Formula: see text] group as well, of the corresponding rank.
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Tamar Bar-On (2024) studied this question.
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