  We prove that the class of closed subgroups of free profinite monoids is
precisely the class of projective profinite groups. In particular, the
profinite groups associated to minimal symbolic dynamical systems by Almeida
are projective. Our result answers a question raised by Lubotzky during the
lecture of Almeida at the Fields Workshop on Profinite Groups and Applications,
Carleton University, August 2005. We also prove that any finite subsemigroup of
a free profinite monoid consists of idempotents.
