撤稿声明: 被撤回的出版物: Linear Representations of Hyperelliptic Mapping Class Groups (Retraction of Vol 71, Pg 221, 2022) (Retracted article. See vol. 71, pg. 221, 2022)

Let p: S -> S-g be a finite G-covering of a closed surface of genus g >= 1, and let B be 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 subgroups 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)(t), which is a subgroup of Mod(S-g \ B) associated with a given hyperelliptic involution. on S-g. We extend some previous results on virtual linear representations of the mapping class group to hyperelliptic mapping class groups. 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 with an unramified G-covering S -> S-2, thus providing a counterexample to the genus 2 case of the Putman-Wieland conjecture.

Automated summary

This paper studies the virtual linear representations of the mapping class group for finite G-coverings, and examines their restrictions on the hyperelliptic mapping class group. It is shown that there exist nontrivial finite orbits for the virtual linear representations of the hyperelliptic mapping class groups, providing a counterexample to the genus 2 case of the Putman-Wieland conjecture.