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On power groups and embedding theorems for relatively free profinite monoids

Published online by Cambridge University Press:  24 February 2005

Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Wien, Austria. e-mail:
School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada. e-mail:


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.

Research Article
2005 Cambridge Philosophical Society

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