Maximal subgroups of the minimal ideal of a free profinite monoid are free

Abstract

We answer a question of Margolis from 1997 by establishing that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. More generally, if H is variety of finite groups closed under extension and containing Z/pZ for infinitely may primes p, the corresponding result holds for free pro-H monoids.

Profinite groups associated to sofic shifts are free

“…In this case, Almeida and Volkov established early on that the corresponding maximal subgroup is free procyclic [10]. In this paper we show that the maximal subgroup associated to a non-minimal irreducible sofic shift is a free profinite group of countable rank, thereby generalizing the result for the minimal ideal [33]. An interesting feature of the proof is the crucial role played by the invariance of the subgroup under conjugacy of dynamical systems.…”

“…Since the full shift is an irreducible sofic shift, an immediate corollary is the main result of [33] (although the proof of that result is simply a specialization of the current proof).…”

“…The fifth section states our main result and proves it modulo a technical lemma. The following section proves the technical lemma, which is based on the argument of [33]. Other results from [10] about the minimal ideal of the free profinite semigroup are generalized to the minimal J -class associated to arbitrary irreducible sofic subshifts.…”

“…Margolis also asked at this time whether all maximal subgroups were projective profinite groups and whether the maximal subgroup of the minimal ideal was a free profinite group. The first question was answered in the positive by Rhodes and the second author [29], whereas the second was answered positively by the second author [33].…”

Presentations of Schützenberger groups of minimal subshifts

“…Rhodes and Steinberg [25] proved that the closed subgroups of free profinite semigroups are precisely the projective profinite groups. Without using ideas from symbolic dynamics, Steinberg proved that the Schützenberger group of the minimal ideal of the free profinite semigroup over a finite alphabet with at least two letters is a free profinite group with infinite countable rank [28]. The same result holds for the Schützenberger group of the regular J -class associated to a non-periodic irreducible sofic subshift [9]; the proof is based on the techniques of [28] and on the conjugacy invariance of the group for arbitrary subshifts [8].…”