To each irreducible subshift over a finite alphabet, one may naturally associate a regular J-class of the free profinite semigroup on the same alphabet, and thus a well-defined abstract profinite group, which is called the Schützenberger group of the subshift. In this talk, we provide a geometric interpretation of this group in the case of minimal subshifts. Indeed, it can be obtained as a natural inverse limit of profinite completions of homotopy groups of graphs combinatorially associated with the minimal subshift, namely a suitable variant of its Rauzy graphs.
(This is joint work with Alfredo Costa.)