## Algebraic & Geometric Topology

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

#### 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 $Bn$, 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 $Bn−1$. 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 $ℤ2$ 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
Revised: 15 November 2001
Accepted: 16 November 2001
First available in Project Euclid: 21 December 2017

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

#### 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

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