Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2

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.

25 pages, 4 figures

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• On an action of the braid group B_{2g+2} on the free group F_{2g}
• The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence
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