Presentations of Schutzenberger groups of minimal subshifts

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 Schützenberger 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 Schützenberger 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.

arXiv commentTo appear in Israel Journal of Mathematics. Corrects metadata field (abstract) of version of 27 Feb 2013

References in corpus (2)

• Closed subgroups of free profinite monoids are projective profinite groups
• Profinite Groups Associated to Sofic Shifts are Free

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• Profinite congruences and unary algebras
• A profinite approach to complete bifix decodings of recurrent languages
• Obstructions to return preservation for episturmian morphisms