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 $B_n$, 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.