On Burau representations at roots of unity

TL;DR

This paper investigates the properties of Burau representations at roots of unity, proving trivial intersections for certain subgroups and classifying the images of braid groups on 3 strands, with implications for conjectures in the field.

Contribution

It establishes new results on the structure of subgroups generated by powers of braid generators and classifies the images of Burau representations at roots of unity.

Findings

Infinite intersections are trivial for even k.

Image of braid group on 3 strands is finite or a finite extension of a triangle group.

Supports conjectures of Squier on kernels of Burau representations.

Abstract

We consider subgroups of the braid groups which are generated by $k$-th powers of the standard generators and prove that any infinite intersection (with even $k$) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau's representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension of a triangle group, by using geometric methods.