Mathematical Research Letters

Volume 29 (2022)

Number 3

Big pure mapping class groups are never perfect

Pages: 691 – 726



George Domat (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)

Ryan Dickmann (Department of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)


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

Main article by George Domat; appendix by Ryan Dickmann and George Domat.