The question of whether the braid groups () 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 [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 and unfaithful for (the case is still unsettled).
In a previous paper [Invent. Math. 142, No. 3, 451-486 (2000; Zbl 0988.20023)], the author defined another representation , where is a free module of rank over a ring , and proved that it is faithful for . S. J. Bigelow [J. Am. Math. Soc. 14, No. 2, 471-486 (2001; Zbl 0988.20021)] showed, using a topological argument, that is faithful for all .
In the present paper, the author exploits combinatorial properties of the action of on to give a completely different proof that is faithful, and hence that all braid groups are linear.