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.

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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

Abstract