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Braid groups are linear. (English) Zbl 1020.20025
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 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).
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 \(n=4\). 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 \(\text{GL}(V)\) to give a completely different proof that \(\rho\) is faithful, and hence that all braid groups are linear.

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