Geodesic
The mapping class group of a genus two surface is linear
Bigelow, Stephen1 ; Budney, Ryan2
1Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3010, Australia
2Department of Mathematics, Cornell University, Ithaca NY 14853-4201, USA
Algebraic and Geometric Topology, Tome 1 (2001) no. 2, pp. 699-708
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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 Bn−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 ℤ2 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.

DOI : 10.2140/agt.2001.1.699
Keywords: mapping class group, braid group, linear, representation

Bigelow, Stephen  1   ; Budney, Ryan  2

1 Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3010, Australia
2 Department of Mathematics, Cornell University, Ithaca NY 14853-4201, USA 
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Bigelow, Stephen; Budney, Ryan. The mapping class group of a genus two surface is linear. Algebraic and Geometric Topology, Tome 1 (2001) no. 2, pp. 699-708. doi: 10.2140/agt.2001.1.699

[1] S J Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471

[2] J S Birman, H M Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. $(2)$ 97 (1973) 424

[3] H M Farkas, I Kra, Riemann surfaces, Graduate Texts in Mathematics 71, Springer (1992)

[4] M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer (1976)

[5] S P Kerckhoff, The Nielsen realization problem, Ann. of Math. $(2)$ 117 (1983) 235

[6] M Korkmaz, On the linearity of certain mapping class groups, Turkish J. Math. 24 (2000) 367

[7] D Krammer, The braid group $B_4$ is linear, Invent. Math. 142 (2000) 451

[8] D Krammer, Braid groups are linear, Ann. of Math. $(2)$ 155 (2002) 131

[9] S Lang, Algebra, Addison-Wesley Publishing Co., Reading, MA (1965)

[10] W B R Lickorish, A finite set of generators for the homeotopy group of a 2–manifold, Proc. Cambridge Philos. Soc. 60 (1964) 769

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