The mapping class group of a genus two surface is linear

By Stephen Bigelow, Ryan Budney. 2000

68 citations recorded by the source corpus.

In this paper we construct a faithful representation of the map- ping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer represen- tation 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. AMS Classication 20F36; 57M07, 20C15

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/525835c53b2335f305033a126b185ad6479a3cf7

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Mapping Class Group from the Viewpoint of Measure Equivalence Theory (2005, background): “(In [7], [58], it is shown that the mapping class groups M2,0 and M0,p for each p are linear, where Mg,p denotes the group of isotopy classes of orientation-preserving self-diffeomorphisms of the closed surface Mg,0 which preserve the set of p marked points on Mg,0.”
Braids: A Survey (2004, background): “explicit representation of M0,0,n constructed in [15] is n ( n− 1 2 2 .”
The asymptotic dimension of a curve graph is finite (2005, background): “he following to get an exact formula for the asdim of the mapping class group of genus-2 surfaces. Let Mod2(Sg,0) denote the elements of Mod(Sg,0) that commute with the hyperelliptic involution, (see [5, 7]). Thinking of Mod2(Sg,0) ⊂ Mod(Sg,0), we see that asdimMod2(Sg,0) ≤ asdimMod(Sg,0). In the ge”