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 (UMR_8243) - 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]

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.

Domaines: Mathématiques [math]; Informatique [cs]

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, 2017, 37 (3), pp.693-715. ⟨10.1017/etds.2015.74⟩. ⟨hal-01367712⟩