# Braid groups are linear

### Abstract

In a previous work [11], the author considered a representation of the braid group $\rho\colon B_n \to \mathrm{GL}_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}]) (m = n(n-1)/2)$, 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.

Daan Krammer