On power groups and embedding theorems for relatively free profinite monoids

Abstract

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