## Geometry & Topology

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

Stephen Bigelow

#### Abstract

The Burau representation is a natural action of the braid group $Bn$ on the free $ℤ[t,t−1]$–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 $n≥9$ and Long and Paton improved on Moody’s techniques to bring this down to $n≥6$. 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 $n≥5$.

#### 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

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

#### References

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• D D Long, M Paton, The Burau representation is not faithful for $n\geq 6$, Topology, 32 (1993) 439–447
• John Atwell Moody, The Burau representation of the braid group $B_n$ is unfaithful for large $n$, Bull. Amer. Math. Soc. 25 (1991) 379–384