Matrix representations of Artin groups

Abstract:

We define matrix representations of Artin groups over a $2$-variable Laurent-polynomial ring and show that in the rank 2 case, the representations are faithful. In the special case of Artin’s braid group, our representation is a version of the Burau representation and our faithfulness theorem is a generalization of the well-known fact that the Burau representation of ${B_3}$ is faithful.