Sturmian morphisms,the braid group B4, Christoffel words and bases of F2 |
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Authors: | Christian Kassel Christophe Reutenauer |
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Affiliation: | (1) Institut de Recherche Mathématique Avancée, CNRS – Université Louis Pasteur, 7 rue René Descartes, 67084 Strasbourg, Cedex, France;(2) Mathématiques, Université du Québec à Montréal, Montréal, CP 8888, succ. Centre Ville, Canada, H3C 3P8 |
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Abstract: | We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F 2) of automorphisms of the rank two free group F 2 and show that it can be realized as a monoid in the group B 4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F 2 lifting any given basis of the free abelian group Z 2. We further give an algorithm allowing to decide whether two elements of F 2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15 |
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Keywords: | Free group Sturmian morphism Braid group Symbolic dynamics Christoffel word |
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