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.

Keywords

mapping class group, braid group, linear, representation

Mathematical Subject Classification 2000

Primary: 20F36

Secondary: 57M07, 20C15

References

Forward citations

Publication

Received: 2 August 2001

Revised: 15 November 2001

Accepted: 16 November 2001

Published: 22 November 2001

Authors

Stephen Bigelow
Department of Mathematics and Statistics University of Melbourne Parkville Victoria, 3010 Australia

Ryan D Budney
Department of Mathematics Cornell University Ithaca NY 14853-4201 USA

Volume 1, issue 2 (2001)

Stephen Bigelow and Ryan D Budney