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arXiv · 1305.0127

The finite index basis property

Valérie Berthé·Clelia De Felice·Francesco Dolce·Julien Leroy·Dominique Perrin·Christophe Reutenauer·Giuseppina Rindone
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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, namemly the class of tree sets. We prove as a main result that for a uniformly recurrent tree set $F$, a finite bifix code $X$ on the alphabet $A$ is $F$-maximal of $F$-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, Christophe Reutenauer, Giuseppina Rindone. 2013-05-01. The finite index basis property. https://arxiv.org/abs/1305.0127

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