Linear representations of hyperelliptic mapping class groups

Abstract

Let p : S → S g be a finite G-covering of a closed surface of genus g ≥ 1 and let B its branch locus. To this data, it is associated a representation of a finite index subgroup of the mapping class group Mod(S g B) in the centralizer of the group G in the symplectic group Sp(H 1 (S, Q)). They are called virtual linear representations of the mapping class group and are related, via a conjecture of Putman and Wieland, to a question of Kirby and Ivanov on the abelianization of finite index subgroup of the mapping class group. The purpose of this paper is to study the restriction of such representations to the hyperelliptic mapping class group Mod(S g , B) ι , which is a subgroup of Mod(S g B) associated to a given hyperelliptic involution ι on S g . We extend to hyperelliptic mapping class groups some previous results on virtual linear representations of the mapping class group. We then show that, for all g ≥ 1, there are virtual linear representations of hyperelliptic mapping class groups with nontrivial finite orbits. In particular, we show that there is such a representation associated to an unramified G-covering S → S 2 , thus providing a counterexample to the genus 2 case of the Putman-Wieland conjecture. A CONJECTURE BY PUTMAN AND WIELANDLet S g be an oriented closed surface of genus g and let P 1 , P 2 , P 3 , . . . be a sequence in S g of pairwise distinct points. We put S g,n := S g {P 1 , . . . , P n }. We always assume that S g,n has negative Euler characteristic: 2g − 2 + n > 0. We abbreviate π 1 (S g,n , P n+1 ) by Π g,n , denote by Mod(S g,n ) the mapping class group of self-homeomorphisms of S g,n and by PMod(S g,n ) the subgroup consisting of mapping classes of self-homeomorphisms which preserve the order of the punctures. Let us recall that these groups act by outer automorphisms on Π g,n and PMod(S g,n ) (if n > 0) acts in a conventional fashion on Π g,n−1 . A fundamental open question on the mapping class group of a surface is the following (cf. Problem 2.11.A in [12] and Problem 7 in [11]): Problem 1.1. Assume g ≥ 3 and n ≥ 0. Is it true that H 1 (Γ) = 0 for every finite index subgroup Γ of Mod(S g,n )?A positive answer has been given for all finite index subgroups of Mod(S g,n ) containing the Johnson subgroup, i.e. the normal subgroup of Mod(S g,n ) generated by Dehn twists about separating simple closed curves (cf. [2] and [17]). More recently, Ershov and He (cf. [6]) showed that this question has a positive answer also for finite index subgroups