Profinite groups associated with weakly primitive substitutions

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.