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Braid groups are linear. (English) Zbl 1020.20025

The question of whether the braid groups ${B}_{n}$ ($n\ge 2$) 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 $n\ge 9$ [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 $n\le 3$ and unfaithful for $n\ge 5$ (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 $\rho :{B}_{n}\to \text{GL}\left(V\right)$, where $V$ is a free module of rank $n\left(n-1\right)/2$ 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 $\rho$ is faithful for all $n$.

In the present paper, the author exploits combinatorial properties of the action of ${B}_{n}$ on $\text{GL}\left(V\right)$ to give a completely different proof that $\rho$ is faithful, and hence that all braid groups are linear.

MSC:
 20F36 Braid groups; Artin groups 57M07 Topological methods in group theory 20C15 Ordinary representations and characters of groups