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Mathematical Research Letters
Volume 29 (2022)
Number 3
Big pure mapping class groups are never perfect
Pages: 691 – 726
DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n3.a4
Authors
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 $\oplus_{2^{\aleph_0}} \:\mathbb{Q}$. 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 $\oplus_{2^{\aleph_0}} \:\mathbb{Z}$ as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to $\mathbb{Q}$.
G. Domat was partially supported by NSF DMS-1607236, NSF DMS-1840190, and NSF DMS-1246989.
Received 12 August 2020
Accepted 11 April 2021
Published 30 November 2022