Gassner and Burau representations over ${\mathbb{Z}}_p$-modules

We study two classical representations of Artin’s braid group and their modulo $p$ reductions. We use topological methods to show that the Gassner representation ${\tau }_n:B_n\to {\text{GL}}_n(\mathbb{Z}[t_1^{\pm 1},\dots ,t_n^{\pm 1}])$ is faithful for all $n$, and furthermore that it is faithful modulo $p$ for all integers $p>1$. We then give a novel proof that the Burau representation of $B_3$ is faithful modulo $p$ for all $p>1$, and suggest applications to the modulo $p$ Burau representation for higher braid groups.