2005, VOLUME 11, NUMBER 3, PAGES 13-48

Profinite groups associated with weakly primitive substitutions

J. Almeida


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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.

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Last modified: September 14, 2005