Big pure mapping class groups are never perfect
Abstract
We show that the closure of the compactly supported mapping class group of an infinite type surface is not perfect and that its abelianization contains a direct summand isomorphic to an uncountable direct sum of rationals. We also extend this to the Torelli group and show that in the case of surfaces with infinite genus the abelianization of the Torelli group contains an indivisible copy of an uncountable free abelian group as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to the rationals.
Cite
@article{arxiv.2007.14929,
title = {Big pure mapping class groups are never perfect},
author = {George Domat and Ryan Dickmann},
journal= {arXiv preprint arXiv:2007.14929},
year = {2021}
}
Comments
v2: Updated to include referees' comments, which includes a new application in Corollary F. To appear in Mathematical Research Letters. 36 pages, 4 figures. Primary article by George Domat with an appendix by Ryan Dickmann and George Domat