Following a correspondence due to Almeida, every uniformly recurrent language determines a maximal subgroup of the free profinite monoid in which it lives, known as the Schützenberger group of the language. These groups are projective profinite groups, but interestingly, they are not always free. Several instances of this have been known early on, with the language of the Thue-Morse substitution being a prime example. In the case of uniformly recurrent languages determined by primitive substitutions, the Schützenberger group admits a profinite presentation of a very special kind (Almeida and Costa, 2013). We make use of this result to study freeness of Schützenberger groups, both absolute and relative. While many aspects of this question remain widely open, we obtain some effective ways to test for absolute freeness of these groups. As an application, we give an example of a primitive invertible substitution whose Schützenberger group is not free (this constitutes a counterexample to a statement put forward by Almeida in 2005). We also show how in some cases, relative invertibility imposes a lower bound (so to speak) on relative freeness.
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Semigroups, Automata and Languages