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Braid groups are linear. (English) Zbl 1020.20025

The question of whether the braid groups B n (n2) are linear is an old one. The most famous representation, the so-called Burau representation, was shown by J. A. Moody not to be faithful for n9 [Bull. Am. Math. Soc., New Ser. 25, No. 2, 379-384 (1991; Zbl 0751.57005)]. It is now known that the Burau representation is faithful for n3 and unfaithful for n5 (the case n=4 is still unsettled).

In a previous paper [Invent. Math. 142, No. 3, 451-486 (2000; Zbl 0988.20023)], the author defined another representation ρ:B n GL(V), where V is a free module of rank n(n-1)/2 over a ring R, and proved that it is faithful for n=4. S. J. Bigelow [J. Am. Math. Soc. 14, No. 2, 471-486 (2001; Zbl 0988.20021)] showed, using a topological argument, that ρ is faithful for all n.

In the present paper, the author exploits combinatorial properties of the action of B n on GL(V) to give a completely different proof that ρ is faithful, and hence that all braid groups are linear.

20F36Braid groups; Artin groups
57M07Topological methods in group theory
20C15Ordinary representations and characters of groups