## 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.

### MSC:

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

### Citations:

Zbl 0751.57005; Zbl 0988.20023; Zbl 0988.20021
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