## Geometry & Topology

- Geom. Topol.
- Volume 3, Number 1 (1999), 397-404.

### The Burau representation is not faithful for $n = 5$

#### Abstract

The Burau representation is a natural action of the braid group ${B}_{n}$ on the free $\mathbb{Z}\left[t,{t}^{-1}\right]$–module of rank $n-1$. It is a longstanding open problem to determine for which values of $n$ this representation is faithful. It is known to be faithful for $n=3$. Moody has shown that it is not faithful for $n\ge 9$ and Long and Paton improved on Moody’s techniques to bring this down to $n\ge 6$. Their construction uses a simple closed curve on the $6$–punctured disc with certain homological properties. In this paper we give such a curve on the $5$–punctured disc, thus proving that the Burau representation is not faithful for $n\ge 5$.

#### Article information

**Source**

Geom. Topol., Volume 3, Number 1 (1999), 397-404.

**Dates**

Received: 21 July 1999

Accepted: 23 November 1999

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.gt/1513883152

**Digital Object Identifier**

doi:10.2140/gt.1999.3.397

**Mathematical Reviews number (MathSciNet)**

MR1725480

**Zentralblatt MATH identifier**

0942.20017

**Subjects**

Primary: 20F36: Braid groups; Artin groups

Secondary: 57M07: Topological methods in group theory 20C99: None of the above, but in this section

**Keywords**

braid group Burau representation

#### Citation

Bigelow, Stephen. The Burau representation is not faithful for $n = 5$. Geom. Topol. 3 (1999), no. 1, 397--404. doi:10.2140/gt.1999.3.397. https://projecteuclid.org/euclid.gt/1513883152