Représentation par des transvections des groupes d'Artin–Tits

Abstract

In a recent article, C. Kassel and C. Reutenauer studied the connection between the 4 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 length n, we deduce a seemingly new representation of the braid group Bn +1 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 for B3 and B4; for other cases, the question of faithfulness remains open.