Profinite groups associated to sofic shifts are free
Alfredo Costa, Benjamin Steinberg · Proceedings of the London Mathematical Society · 2010
We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the minimal ideal of the free profinite semigroup and the J-class associated to an irreducible sofic shift.