## Algebraic & Geometric Topology

- Algebr. Geom. Topol.
- Volume 1, Number 2 (2001), 699-708.

### The mapping class group of a genus two surface is linear

Stephen Bigelow and Ryan Budney

#### Abstract

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence–Krammer representation of the braid group ${B}_{n}$, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful representation of the mapping class group of the $n$–punctured sphere by using the close relationship between this group and ${B}_{n-1}$. We then extend this to a faithful representation of the mapping class group of the genus two surface, using Birman and Hilden’s result that this group is a ${\mathbb{Z}}_{2}$ central extension of the mapping class group of the $6$–punctured sphere. The resulting representation has dimension sixty-four and will be described explicitly. In closing we will remark on subgroups of mapping class groups which can be shown to be linear using similar techniques.

#### Article information

**Source**

Algebr. Geom. Topol., Volume 1, Number 2 (2001), 699-708.

**Dates**

Received: 2 August 2001

Revised: 15 November 2001

Accepted: 16 November 2001

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.agt/1513882644

**Digital Object Identifier**

doi:10.2140/agt.2001.1.699

**Mathematical Reviews number (MathSciNet)**

MR1875613

**Zentralblatt MATH identifier**

0999.57020

**Subjects**

Primary: 20F36: Braid groups; Artin groups

Secondary: 57M07: Topological methods in group theory 20C15: Ordinary representations and characters

**Keywords**

mapping class group braid group linear representation

#### Citation

Bigelow, Stephen; Budney, Ryan. The mapping class group of a genus two surface is linear. Algebr. Geom. Topol. 1 (2001), no. 2, 699--708. doi:10.2140/agt.2001.1.699. https://projecteuclid.org/euclid.agt/1513882644