Notes on hyperelliptic mapping class groups
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. This fills a gap in the proof of Theorem 3.13 in \emph{Linear representations of hyperelliptic mapping class groups} appeared in MMJ. 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.13534
 Bibcode:
 2021arXiv211013534B
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Geometry;
 57K20 (Primary) 14H10 (Secondary)
 EPrint:
 35 pages