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

Godelle, Eddy

Groups Geom. Dyn. 1, No. 2, 111-133 (2007).

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

Keywords:

representations of braid groups; automorphisms of free groups; episturmian morphisms; Artin-Tits groups; affine braid groups

Citations:

Zbl 1150.05042