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 $p > 1$.
Categories: math.GT math.RT