On the Burau Representation of $B_4$ modulo $p$

Abstract

The problem of faithfulness of the (reduced) Burau representation for $n=4$ is known to be equivalent to the problem of whether certain two matrices $A$ and $B$ generate a free group of rank two. It is known that $A^3$ and $B^3$ generate a free group of rank two \cite{9}, \cite{10}, \cite{4}. We prove that they also generate a free group when considered as matrices over the ${\mathbb{Z}}_p[t,t^{-1}]$ for any integer $p>1$.