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 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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- October 2000
- DOI:
- 10.48550/arXiv.math/0010310
- arXiv:
- arXiv:math/0010310
- Bibcode:
- 2000math.....10310B
- Keywords:
-
- Mathematics - Geometric Topology;
- Mathematics - Algebraic Topology;
- Mathematics - Group Theory;
- 20F36;
- 57M07;
- 20C15
- E-Print:
- Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-34.abs.html