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Zbl 1020.20025
Krammer, Daan
Braid groups are linear.
(English)
[J] Ann. Math. (2) 155, No. 1, 131-156 (2002). ISSN 0003-486X; ISSN 1939-0980/e

The question of whether the braid groups $B_n$ ($n\geq 2$) are linear is an old one. The most famous representation, the so-called Burau representation, was shown by {\it J. A. Moody} not to be faithful for $n\geq 9$ [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 $n\leq 3$ and unfaithful for $n\geq 5$ (the case $n=4$ is still unsettled).\par In a previous paper [Invent. Math. 142, No. 3, 451-486 (2000; Zbl 0988.20023)], the author defined another representation $\rho\colon B_n\to\text{GL}(V)$, where $V$ is a free module of rank $n(n-1)/2$ over a ring $R$, and proved that it is faithful for $n=4$. {\it 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 $n$.\par In the present paper, the author exploits combinatorial properties of the action of $B_n$ on $\text{GL}(V)$ to give a completely different proof that $\rho$ is faithful, and hence that all braid groups are linear.
[Carl Droms (Harrisonburg)]
MSC 2000:
*20F36 Braid groups; Artin groups
57M07 Topological methods in group theory
20C15 Ordinary representations and characters of groups

Keywords: braid groups; linear groups; Burau representation; faithful representations

Citations: Zbl 0751.57005; Zbl 0988.20023; Zbl 0988.20021

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