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The mapping class group of a genus two surface is linear

Stephen Bigelow and Ryan D Budney

Algebraic & Geometric Topology 1 (2001) 699–708

arXiv: math.GT/0010310


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 Bn, 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 Bn1. 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 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.

mapping class group, braid group, linear, representation
Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 57M07, 20C15
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Received: 2 August 2001
Revised: 15 November 2001
Accepted: 16 November 2001
Published: 22 November 2001
Stephen Bigelow
Department of Mathematics and Statistics
University of Melbourne
Victoria, 3010
Ryan D Budney
Department of Mathematics
Cornell University
Ithaca NY 14853-4201