Profinite groups associated to sofic shifts are free

Abstract

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 kernel of the free profinite semigroup and the J -class associated to an irreducible sofic shift.

Presentations of Schützenberger groups of minimal subshifts

“…In recent years, several results on closed subgroups of free profinite semigroups have appeared in the literature [3,4,6,25,28,9]. The first author explored a link between symbolic dynamics and free profinite semigroups that allowed him to show, for several classes of maximal subgroups of free profinite semigroups, all associated with minimal subshifts [3,4], that they are free profinite groups.…”

The linear nature of pseudowords

“…We recall the concept of entropy of a pseudoword, first introduced in [9], and applied there in the study of relatively free profinite semigroups. Some further applications were given in [17]. Let V be a pseudovariety containing LSl and A an alphabet with at least two letters.…”

“…This is the definition used in[17]. In [9] the entropy of w is defined as 1 n log |A| qw(n), which equals h(w) log |A| 2 for h(w) as defined here.…”

A geometric interpretation of the Schützenberger group of a minimal subshift

“…The Schützenberger group of the full shift A Z is isomorphic to the maximal subgroups of the minimum ideal of Ω A S and was first identified in [34], with techniques that were later extended to the general sofic case in [17] taking into account the invariance of G(X ) under conjugacy of symbolic dynamical systems [15]. This led to the main result of [17] that G(X ) is a free profinite group with rank ℵ 0 when X is a non-periodic irreducible sofic subshift. 1 From the viewpoint of the structure of the group G(X ), the class of irreducible sofic subshifts is thus quite different from that of substitutive (minimal) subshifts.…”