Algebraic and Geometric Topology 1 (2001), paper no. 34, pages 699-708.

The mapping class group of a genus two surface is linear

Stephen J. Bigelow, Ryan D. Budney


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.

Keywords. Mapping class group, braid group, linear, representation

AMS subject classification. Primary: 20F36. Secondary: 57M07, 20C15.

DOI: 10.2140/agt.2001.1.699

E-print: arXiv:math.GT/0010310

Submitted: 2 August 2001. (Revised: 15 November 2001.) Accepted: 16 November 2001. Published: 22 November 2001.

Notes on file formats

Stephen J. Bigelow, Ryan D. Budney
Department of Mathematics and Statistics, University of Melbourne
Parkville, Victoria, 3010, Australia
and
Department of Mathematics, Cornell University
Ithaca, New York 14853-4201, USA

Email: bigelow@unimelb.edu.au, rybu@math.cornell.edu

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