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.
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Research funded by the European Regional Development Fund, through the programme COMPETE, by the Portuguese Government through Centro de Matemática da Universidade do Porto, Centre for Mathematics of the University of Coimbra, and FCT — Fundação para a Ciência e a Tecnologia, under the projects PEst-C/MAT/UI0144/2011 and PEst-C/MAT/UI0324/2011, and by the FCT project PTDC/MAT/65481/2006, within the framework of the programmes COMPETE and FEDER.
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Almeida, J., Costa, A. Presentations of Schützenberger groups of minimal subshifts. Isr. J. Math. 196, 1–31 (2013). https://doi.org/10.1007/s11856-012-0139-4
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DOI: https://doi.org/10.1007/s11856-012-0139-4