#### Volume 1, issue 2 (2001)

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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
##### 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
##### Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 57M07, 20C15