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The mapping class group of a genus two surface is linear | | Stephen J. Bigelow
; Ryan D. Budney
; | | Date: |
31 Oct 2000 | | Journal: | Algebr. Geom. Topol. 1 (2001) 699-708 | | Subject: | Geometric Topology; Algebraic Topology; Group Theory MSC-class: 20F36, 57M07, 20C15 | math.GT math.AT math.GR | | 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. | | Source: | arXiv, math.GT/0010310 | | Services: | Forum | Review | PDF | Favorites | |
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