#### 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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