Profinite Groups Associated to Sofic Shifts are Free

TL;DR

This paper proves that the maximal subgroup of the free profinite semigroup linked to an irreducible sofic shift is a free profinite group, extending previous results and exploring related algebraic structures.

Contribution

It generalizes earlier findings by showing the maximal subgroup is free for a broader class of shifts and establishes new analogies in profinite semigroup theory.

Findings

Maximal subgroup of free profinite semigroup is free for irreducible sofic shifts

Results extend previous work on full shifts to more general shifts

Identifies analogies between kernel of free profinite semigroup and $ ext{J}$-class

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.