Braid groups are linear
Group Theory
2007-05-23 v1 Geometric Topology
Abstract
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.
Cite
@article{arxiv.math/0405198,
title = {Braid groups are linear},
author = {Daan Krammer},
journal= {arXiv preprint arXiv:math/0405198},
year = {2007}
}
Comments
26 pages, published version