The mapping class group of a genus two surface is linear

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.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-34.abs.htm

A family of representations of braid groups on surfaces

“…Bardakov [2005] applied this linearity to show that the braid groups of the sphere and projective plane are linear. Bigelow and Budney [2001] proved using the LawrenceKrammer-Bigelow representation and a suitable branched covering that the mapping class group of genus 2 surface has a faithful linear representation. However, Paoluzzi and Paris showed that there is a difference between V and H 2 (B n,2 )(D) as a ‫[ޚ‬q ±1 , t ±1 ]-module for n ≥ 3 and a found basis for a ‫[ޚ‬q ±1 , t ±1 ]-module H 2 (B n,2 (D)); so the exact definition of "Lawrence-Krammer-Bigelow representation" became somewhat ambiguous.…”

On the linearity problem for mapping class groups

“…We assume throughout that maps fix boundary components pointwise. Bigelow and Budney [3] and independently Korkmaz [13] recently determined that M 2,0,0 is linear. Korkmaz also showed in [13] that mapping class groups contain very large linear subgroups, namely, the hyperelliptic subgroups.…”

Frattini and related subgroups of mapping class groups

“…⊔ ⊓ Remark 1: Recall that the hyperelliptic Mapping Class Group (which we denote by Γ h g ) is defined to be the subgroup of Γ g consisting of those elements that commute with a fixed hyperelliptic involution. It is pointed out in [6] p. 706, that the arguments used in [6] prove that Γ h g is linear.…”

“…As for the case g ≤ 2, note that Γ 1 and Γ 2 are linear (see [6] for g = 2) and so Platonov's result [26] applies to answer Ivanov's question in the affirmative in these cases for all finitely generated subgroups. On the other hand, for g ≥ 3 the Mapping Class Group is not known to be linear and no other technique for answering Ivanov's question was known.…”