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Return words of linear involutions and fundamental groups

Published online by Cambridge University Press:  28 December 2015

VALÉRIE BERTHÉ ,
VINCENT DELECROIX ,
FRANCESCO DOLCE ,
DOMINIQUE PERRIN ,
CHRISTOPHE REUTENAUER  and
GIUSEPPINA RINDONE
VALÉRIE BERTHÉ
Affiliation:
CNRS, LIAFA, Université Paris Diderot, Paris 7 – Case 7014, F-75205 Paris Cedex 13, France
VINCENT DELECROIX
Affiliation:
LaBRI, UMR 5800, Bâtiment A30, 351, cours de la Libération, 33405 Talence Cedex, France
FRANCESCO DOLCE
Affiliation:
Université Paris-Est Marne-la-Vallée, Laboratoire d’informatique Gaspard-Monge, UMR 8049 CNRS, 5 Bd Descartes, Champs-sur-Marne, F-77454 Marne-la-Vallée Cedex 2, France email dominique.perrin@esiee.fr
DOMINIQUE PERRIN
Affiliation:
Université Paris-Est Marne-la-Vallée, Laboratoire d’informatique Gaspard-Monge, UMR 8049 CNRS, 5 Bd Descartes, Champs-sur-Marne, F-77454 Marne-la-Vallée Cedex 2, France email dominique.perrin@esiee.fr
CHRISTOPHE REUTENAUER
Affiliation:
Département de mathématiques, Université du Québec à Montréal, C.P. 8888, Succursale Centre-Ville Montréal, Québec H3C 3P8, Canada
GIUSEPPINA RINDONE
Affiliation:
Université Paris-Est Marne-la-Vallée, Laboratoire d’informatique Gaspard-Monge, UMR 8049 CNRS, 5 Bd Descartes, Champs-sur-Marne, F-77454 Marne-la-Vallée Cedex 2, France email dominique.perrin@esiee.fr
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Abstract

We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions 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 $G$ .

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 3 , May 2017 , pp. 693 - 715
Copyright
© Cambridge University Press, 2015 

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