On power groups and embedding theorems for relatively free profinite monoids

We determine those pseudovarieties of groups H for which the power monoids P(G), ranging over all groups G in H, satisfy the same profinite identities (i.e. pseudoidentities) as all semidirect products of J-trivial rnonoids by groups in H. That is, in the language of finite monoid theory, we characterize all solutions to the pseudovariety equation PH = J*H. The characterization is in terms of the geometry of the Cayley graphs of the free pro-H groups as well as in terms of the pro-H topology of a finitely generated free group.