The finite index basis property

Abstract : We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namely the class of tree sets. We prove as a main result that for a uniformly recurrent tree set S, a finite bifix code X on the alphabet A is S-maximal of S-degree d if and only if it is the basis of a subgroup of index d of the free group on A.
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Valérie Berthé, Clelia de Felice, Francesco Dolce, Julien Leroy, Dominique Perrin, et al.. The finite index basis property. Journal of Pure and Applied Algebra, Elsevier, 2015, ⟨10.1016/j.jpaa.2014.09.014⟩. ⟨hal-01367681⟩

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