%0 Journal Article
%T The mapping class group of a genus two surface is linear
%A Stephen J. Bigelow
%A Ryan D. Budney
%J Mathematics
%D 2000
%I arXiv
%R 10.2140/agt.2001.1.699
%X 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.
%U http://arxiv.org/abs/math/0010310v3