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

TL;DR

This paper proves that the matrices generating a free group in the Burau representation of $B_4$ over integers also generate a free group over the finite field $ ext{Z}_p$, for any integer $p > 1$, advancing understanding of faithfulness.

Contribution

It extends the known freeness of matrices $A^3$ and $B^3$ to matrices over $ ext{Z}_p[t,t^{-1}]$ for all integers $p > 1$, providing new insights into the Burau representation.

Findings

Matrices $A^3$ and $B^3$ generate a free group over $ ext{Z}_p[t,t^{-1}]$ for all $p > 1$

The faithfulness problem of the Burau representation relates to free group generation

The result applies to the reduced Burau representation at $n=4$

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 $p > 1$.