Algebraic & Geometric Topology

The mapping class group of a genus two surface is linear

Stephen Bigelow and Ryan 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 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.

Article information

Source
Algebr. Geom. Topol., Volume 1, Number 2 (2001), 699-708.

Dates
Received: 2 August 2001
Revised: 15 November 2001
Accepted: 16 November 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882644

Digital Object Identifier
doi:10.2140/agt.2001.1.699

Mathematical Reviews number (MathSciNet)
MR1875613

Zentralblatt MATH identifier
0999.57020

Subjects
Primary: 20F36: Braid groups; Artin groups
Secondary: 57M07: Topological methods in group theory 20C15: Ordinary representations and characters

Keywords
mapping class group braid group linear representation

Citation

Bigelow, Stephen; Budney, Ryan. The mapping class group of a genus two surface is linear. Algebr. Geom. Topol. 1 (2001), no. 2, 699--708. doi:10.2140/agt.2001.1.699. https://projecteuclid.org/euclid.agt/1513882644


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References

  • Stephen J. Bigelow, Braid Groups are Linear. J. Amer. Math. Soc. 14 (2001), no.2, 471–486.
  • Joan S. Birman, Hugh M. Hilden, On Isotopies of Homeomorphisms of Riemann Surfaces, Ann. Math. (2) 97 (1973), 424–439.
  • H. M. Farkas, I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, 71. Springer-Verlag, New York, 1992.
  • Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976.
  • Steven P. Kerckhoff, The Nielsen Realization Problem. Ann. Math. (2) 117 (1983), 235–265.
  • Mustafa Korkmaz, On the linearity of certain mapping class groups, Turkish J. Math. 24 (2000), no. 4, 367–371.
  • Daan Krammer, The braid group ${B}_4$ is linear, Invent. Math. 142 (2000), no.3, 451–486.
  • Daan Krammer, Braid groups are linear, Ann. of Math. (2), to appear.
  • Serge Lang, Algebra, 3rd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1993.
  • W.B.R. Lickorish, A Finite Set of Generators for the Homeotopy Group of a 2-Manifold. Proc. Cambridge Philos. Soc. 60 (1964), 768–778.