Presentations of Schutzenberger groups of minimal subshifts

TL;DR

This paper explores the structure of Schützenberger groups associated with minimal subshifts, proving a conjecture for substitution subshifts and analyzing the Prouhet-Thue-Morse case to determine its group properties.

Contribution

It establishes the conjecture for substitution subshifts and provides a detailed analysis of the Prouhet-Thue-Morse group's presentation and properties.

Findings

Conjecture proven for all non-periodic substitution subshifts.

Prouhet-Thue-Morse Schützenberger group has rank three.

The group is non-free relative to any pseudovariety of groups.

Abstract

In previous work, the first author established a natural bijection between minimal subshifts and maximal regular J-classes of free profinite semigroups. In this paper, the Sch\"utzenberger groups of such J-classes are investigated, in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for all non-periodic minimal subshifts associated with substitutions. It entails that it is decidable whether a finite group is a quotient of such a profinite group. As a further application, the Sch\"utzenberger group of the J-class corresponding to the Prouhet-Thue-Morse subshift is shown to admit a somewhat simpler presentation, from which it follows that it has rank three, and that it is non-free relatively to any pseudovariety of groups.