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 LawrenceKrammer 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 npunctured sphere by using the close relationship between this group and B_{n1}. 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 6punctured sphere. The resulting representation has dimension sixtyfour 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2000
 arXiv:
 arXiv:math/0010310
 Bibcode:
 2000math.....10310B
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Topology;
 Mathematics  Group Theory;
 20F36;
 57M07;
 20C15
 EPrint:
 Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt134.abs.html