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

TL;DR

This paper proves the faithfulness of the Gassner and Burau representations of braid groups over finite fields, using topological methods, and explores their properties modulo prime numbers.

Contribution

It establishes the faithfulness of the Gassner representation over olds and primes, and provides a new proof for the Burau representation of B3 modulo p.

Findings

Gassner representation is faithful over olds and modulo p for all p>1

Burau representation of B3 is faithful modulo p for all p>1

Topological methods are effective in analyzing braid group representations

Abstract

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 $\tau_n: B_n\to\text{GL}_n(\mathbb{Z}[t_1^{\pm 1}, \ldots, t_n^{\pm 1}])$ 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.