Normal subgroups of big mapping class groups

Calegari, Danny; Chen, Lvzhou

doi:10.1090/btran/108

Normal subgroups of big mapping class groups
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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