Braid groups are linear

Details Braid groups are linear Braid groups are linear

2004-05-11

arxiv.org/abs/math/0405198v1

Ann. of Math. (2), Vol. 155 (2002), no. 1, 131--156

Mathematics

Group Theory

26 pages, published version

Scientific paper

In a previous work [11], the author considered a representation of the braid group \rho: B_n\to GL_m(\Bbb Z[q^{\pm 1},t^{\pm 1}]) (m=n(n-1)/2), and proved it to be faithful for n=4. Bigelow [3] then proved the same representation to be faithful for all n by a beautiful topological argument. The present paper gives a different proof of the faithfulness for all n. We establish a relation between the Charney length in the braid group and exponents of t. A certain B_n-invariant subset of the module is constructed whose properties resemble those of convex cones. We relate line segments in this set with the Thurston normal form of a braid.

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