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Representation of Artin-Tits groups by transvections. (Repr√©sentation 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

Citations:

Zbl 1150.05042
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