Notes on hyperelliptic mapping class groups

TL;DR

This paper systematically studies hyperelliptic mapping class groups, exploring their structure, representations, and profinite completions, providing counterexamples to existing conjectures and extending the congruence subgroup property.

Contribution

It offers a comprehensive analysis of hyperelliptic mapping class groups, including their centralizers, representations, and profinite properties, and presents counterexamples to conjectures.

Findings

Counterexample to genus 2 case of Putman-Wieland conjecture

Extension of the congruence subgroup property to hyperelliptic groups

Determination of centralizers of multitwists in profinite completions

Abstract

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.