## Normal subgroups of big mapping class groups

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Danny Calegari and Lvzhou Chen
**HTML**| PDF - 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

- Santana Afton, Danny Calegari, Lvzhou Chen, and Rylee Alanza Lyman,
*Nielsen realization for infinite-type surfaces*, Proc. Amer. Math. Soc.**149**(2021), no. 4, 1791–1799. MR**4242332**, DOI 10.1090/proc/15316 - R. D. Anderson,
*The algebraic simplicity of certain groups of homeomorphisms*, Amer. J. Math.**80**(1958), 955–963. MR**98145**, DOI 10.2307/2372842 - Javier Aramayona, Priyam Patel, and Nicholas G. Vlamis,
*The first integral cohomology of pure mapping class groups*, Int. Math. Res. Not. IMRN**22**(2020), 8973–8996. MR**4216709**, DOI 10.1093/imrn/rnaa229 - Javier Aramayona and Nicholas G. Vlamis,
*Big mapping class groups: an overview*, In the tradition of Thurston—geometry and topology, Springer, Cham, [2020] ©2020, pp. 459–496. MR**4264585**, DOI 10.1007/978-3-030-55928-1_{1}2 - Philip Boyland,
*Topological methods in surface dynamics*, Topology Appl.**58**(1994), no. 3, 223–298. MR**1288300**, DOI 10.1016/0166-8641(94)00147-2 - Danny Calegari and Lvzhou Chen,
*Big mapping class groups and rigidity of the simple circle*, Ergodic Theory Dynam. Systems**41**(2021), no. 7, 1961–1987. MR**4266358**, DOI 10.1017/etds.2020.43 - Thomas Church and Andrew Putman,
*Generating the Johnson filtration*, Geom. Topol.**19**(2015), no. 4, 2217–2255. MR**3375526**, DOI 10.2140/gt.2015.19.2217 - Benson Farb and Dan Margalit,
*A primer on mapping class groups*, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR**2850125** - Louis Funar,
*On power subgroups of mapping class groups*, J. Gökova Geom. Topol. GGT**8**(2014), 14–34. MR**3310570** - B. von Kerékjártó,
*Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche*, Math. Ann.**80**(1919), no. 1, 36–38 (German). MR**1511945**, DOI 10.1007/BF01463232 - Mustafa Korkmaz,
*Generating the surface mapping class group by two elements*, Trans. Amer. Math. Soc.**357**(2005), no. 8, 3299–3310. MR**2135748**, DOI 10.1090/S0002-9947-04-03605-0 - Justin Lanier and Dan Margalit,
*Normal generators for mapping class groups are abundant*, Comment. Math. Helv.**97**(2022), no. 1, 1–59. MR**4410724**, DOI 10.4171/cmh/526 - F. Luo,
*Torsion elements in the mapping class group of a surface*, Preprint, arXiv:math/0004048, 2000. - Justin Malestein and Jing Tao,
*Self-similar surfaces: involutions and perfection*, Preprint, arXiv:2106.03681, 2021. - Priyam Patel and Nicholas G. Vlamis,
*Algebraic and topological properties of big mapping class groups*, Algebr. Geom. Topol.**18**(2018), no. 7, 4109–4142. MR**3892241**, DOI 10.2140/agt.2018.18.4109 - Nicholas G. Vlamis,
*Three perfect mapping class groups*, New York J. Math.**27**(2021), 468–474. MR**4226155**

## Additional Information

**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
- © Copyright 2022 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B
**9**(2022), 957-976 - MSC (2020): Primary 57K20, 20F05, 20E07; Secondary 37E30, 20J06
- DOI: https://doi.org/10.1090/btran/108
- MathSciNet review: 4498366