# Transactions of the American Mathematical Society Series B

Published by the American Mathematical Society since 2014, this gold open access electronic-only journal is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 2330-0000

The 2020 MCQ for Transactions of the American Mathematical Society Series B is 1.73.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Normal subgroups of big mapping class groupsHTML articles powered by AMS MathViewer

by Danny Calegari and Lvzhou Chen
Trans. Amer. Math. Soc. Ser. B 9 (2022), 957-976

## Abstract:

Let \$S\$ be a surface and let \$\operatorname {Mod}(S,K)\$ be the mapping class group of \$S\$ permuting a Cantor subset \$K \subset S\$. We prove two structure theorems for normal subgroups of \$\operatorname {Mod}(S,K)\$.

(Purity:) if \$S\$ has finite type, every normal subgroup of \$\operatorname {Mod}(S,K)\$ either contains the kernel of the forgetful map to the mapping class group of \$S\$, or it is ‘pure’ — i.e. it fixes the Cantor set pointwise.

(Inertia:) for any \$n\$ element subset \$Q\$ of the Cantor set, there is a forgetful map from the pure subgroup \$\operatorname {PMod}(S,K)\$ of \$\operatorname {Mod}(S,K)\$ to the mapping class group of \$(S,Q)\$ fixing \$Q\$ pointwise. If \$N\$ is a normal subgroup of \$\operatorname {Mod}(S,K)\$ contained in \$\operatorname {PMod}(S,K)\$, its image \$N_Q\$ is likewise normal. We characterize exactly which finite-type normal subgroups \$N_Q\$ arise this way.

Several applications and numerous examples are also given.

References
Similar Articles
• Danny Calegari
• Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois, 60637
• MR Author ID: 605373
• Email: dannyc@math.uchicago.edu
• Lvzhou Chen
• Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas, 78712; and Department of Mathematics, Purdue University, West Lafayette, Indiana, 47907
• MR Author ID: 1271475
• ORCID: 0000-0001-9039-9745
• Email: lvzhou.chen@math.utexas.edu
• Received by editor(s): November 1, 2021
• Received by editor(s) in revised form: January 3, 2022
• Published electronically: October 19, 2022