Profinite groups associated with weakly primitive substitutions

Abstract

A uniformly recurrent pseudoword is an element of a free profinite semigroup in which every finite factor appears in every sufficiently long finite factor. An alternative characterization is as a pseudoword that is a factor of all its infinite factors, i.e., one that lies in a \(\mathcal{J}\)-class with only finite words strictly \(\mathcal{J}\)-above it. Such a \(\mathcal{J}\)-class is regular, and therefore it has an associated profinite group, namely, any of its maximal subgroups. One way to produce such \(\mathcal{J}\)-classes is to iterate finite weakly primitive substitutions. This paper is a contribution to the computation of the profinite group associated with the \(\mathcal{J}\)-class that is generated by the infinite iteration of a finite weakly primitive substitution. The main result implies that the group is a free profinite group provided the substitution induced on the free group on the letters that appear in the images of all of its sufficiently long iterates is invertible.