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Braid groups are linear

Daan Krammer
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Abstract

In a previous work [11], the author considered a representation of the braid group ρ: 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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Daan Krammer. 2004-05-11. Braid groups are linear. https://arxiv.org/abs/math/0405198

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