The mapping class group of a genus two surface is linear

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 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.