Représentation par des transvections des groupes d'Artin–Tits

In a recent article, C. Kassel and C. Reutenauer studied the connection between the4 strand braid group and Sturmian morphisms in word combinatorics. The aim of the current work is to extend this approach into a general connection between braid groups (of any index) and episturmian morphisms, a natural generalization of sturmian morphisms. Our key tool consists in associating with every graph a certain finite family of automorphisms of a free group. In the case of a complete graph, we recover some well-known family of episturmian morphisms. Now, considering the path of lengthn, we deduce a seemingly new representation of the braid group BnC1 in Aut.Fn/. By considering some other graphs, we similarly obtain representations of various Artin-Tits groups, in particular some affine braid groups. Our representation is faithful forB3 andB4; for other cases, the question of faithfulness remains open.