Return words of linear involutions and fundamental groups

Valérie Berthé 1 Vincent Delecroix 2 Francesco Dolce 3 Dominique Perrin 4, 3 Christophe Reutenauer 5 Giuseppina Rindone 3
1 IRIF - Institut de Recherche en Informatique Fondamentale
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 LIGM - Laboratoire d'Informatique Gaspard-Monge
4 ESIEE Paris
5 LaCIM - Laboratoire de combinatoire et d'informatique mathématique [Montréal]
Abstract : We investigate the shifts associated with natural codings of linear in-volutions. We deduce from the geometric representation of linear invo-lutions as Poincaré maps of measured foliations a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet A. The set of return words with respect to a subgroup of finite index G of the free group on A is also proved to be a symmetric basis of G.
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Valérie Berthé, Vincent Delecroix, Francesco Dolce, Dominique Perrin, Christophe Reutenauer, et al.. Return words of linear involutions and fundamental groups. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2017, 37 (3), pp.693-715. ⟨10.1017/etds.2015.74⟩. ⟨hal-01367712⟩

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