# Linear-central filtrations and the image of the Burau representation

## TL;DR

This paper demonstrates that the image of the Burau representation of the braid group is dense in a certain unitary group, providing a strong approximation to a longstanding question about its image and suggesting broader applicability to other representations.

## Contribution

It introduces a new approach using linear-central filtrations to show the density of the Burau representation's image in a unitary group, advancing understanding of its structure.

## Key findings

- Burau representation preserves a Hermitian form
- The image is dense in a unitary group under a natural topology
- Methods may extend to other braid group representations

## Abstract

The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of Birman asks for a description of the image; in this paper we give a "strong approximation" to the answer. Since a 1984 paper of Squier it has been known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group relative to a topology induced by a naturally-occurring filtration. We expect that the methods of the paper should extend to many other representations of the braid group and perhaps ultimately inform the study of knot and link polynomials.