Presentations of Schützenberger groups of minimal subshifts

There is a natural bijection between minimal subshifts over a finite alphabet A and maximal regular J-classes of the profinite semigroup freely generated by A. 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 several types of minimal subshifts associated with substitutions. The Schützenberger subgroup 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 satisfies the conjecture, that it has rank three, and that it is non-free relatively to any pseudovariety of groups. (Joint work with Alfredo Costa, from CMUC/University of Coimbra.)