Automorphismes de groupes libres, tresses et morphismes épisturmiens
Résumé
In a recent article, C. Kassel et C. Reutenauer studied the link betwenn the braid group with 4 strings and sturmian morphisms. The main target of our article is to show that there exists a general connection between braids groups and episturmian morphisms, which are the natural generalization of sturmian morphisms. We introduce the notion of the automorphisms of a free group associated with a graph. In the case of the connected and simply connected graph, we obtain a representation of the braid group, which seems to be new . In the case of the complete graph, we obtain a well-known family of episturmian morphisms. Our method allows us to construct representations of some Artin-Tits groups, and in particular of some affine braid groups. Our representation is faithful for $B_3$ and $B_4$. For the other cases, the question is open. Finally, we deduce from our representation some new results for braid groups.