Faithfulness of the Lawrence representation of braid groups

TL;DR

This paper proves that the Lawrence representation of braid groups is faithful for all parameters m greater than or equal to 2, enhancing understanding of braid group representations.

Contribution

It establishes the faithfulness of the Lawrence representation for all m ≥ 2, extending previous results and connecting it to well-known braid group representations.

Findings

Lawrence representation is faithful for m ≥ 2

Special cases include reduced Burau and Lawrence-Krammer representations

Provides new insights into braid group representations

Abstract

The Lawrence representation $L_{n,m}$ is a family of homological representation of the braid group $B_n$, which specializes to the reduced Burau and the Lawrence-Krammer representation when $m$ is 1 and 2. In this article we show that the Lawrence representation is faithful for $m \geq 2$.