## Representation of Artin-Tits groups by transvections. (Représentation par des transvections des groupes d’Artin-Tits.)(French)Zbl 1151.20031

Summary: In a recent article [Ann. Mat. Pura Appl. (4) 186, No. 2, 317-339 (2007; Zbl 1150.05042)], 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 $$B_{n+1}$$ in $$\operatorname{Aut}(F_n)$$. 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 $$B_3$$ and $$B_4$$; for other cases, the question of faithfulness remains open.

### MSC:

 20F36 Braid groups; Artin groups 20C15 Ordinary representations and characters 68R15 Combinatorics on words 20E05 Free nonabelian groups

Zbl 1150.05042
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