The mapping class group of a genus two surface is linear

Stephen J. Bigelow; Ryan D. Budney

doi:10.2140/agt.2001.1.699

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The mapping class group of a genus two surface is linear

Stephen J. Bigelow·Ryan D. Budney

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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.

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BibTeX RIS

Stephen J. Bigelow, Ryan D. Budney. 2000-10-31. The mapping class group of a genus two surface is linear. https://doi.org/10.2140/agt.2001.1.699

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