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Let $\Sigma$ be a finite-type orientable surface with negative Euler characteristic, and $\mathrm{Mod}(\Sigma)$ denote the mapping class group. What are the finite normal subgroups in $\mathrm{Mod}(\Sigma)$?

For example, when $\Sigma$ is a genus 2 closed surface, then the hyperelliptic involution $\sigma$ lies in the center of $\mathrm{Mod}(\Sigma)$, so it generates a normal subgroup with 2 elements. Are there any other finite normal subgroups, or which one is the maximal one?

For higher genus, does there exist a non-trivial finite normal subgroup?

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2 Answers 2

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I found the following two assertions in Maher and Tiozzo's article Random walks, WPD actions, and the Cremona group. They refer to Ivanov's monograph Subgroups of Teichmüller modular groups. (I don't have have access to the paper right now, so I did not check the statements.)

  • Every finite normal subgroup is contained in the centre. [Iv, Exercise 5, Section 11]
  • If $\mathrm{Mod}(S_{g,n})$ is infinite, then its centre is trivial unless $(g,n) \in \{ (1,0), (1,1), (1,2), (2,0)\}$, in which case it is generated by the hyperelliptic involution. [Iv, Remark 8.15]
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  • $\begingroup$ Just to complete, when is $\mathrm{Mod}(S_{g,n})$ finite? I'd expect it is precisely when $(g,n)\in\{(0,0),(0,1)\}$ and then it is reduced to identity (assuming Mod consists of orientation-preserving mapping classes). By the way, for $(0,2)$ (annulus), the mapping class group is infinite cyclic, so has nontrivial center (but the extended mapping class group is infinite dihedral and has trivial center — also allowing oriented class swapping the components makes it infinite dihedral). $\endgroup$
    – YCor
    Commented May 29 at 8:01
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    $\begingroup$ @YCor - the answer to your question depends on the definition of the mapping class group. There is (a) the group of homeomorphisms modulo isotopy. In this case the finite mapping class groups occur for the data (0, 0), (0, 1), (0, 2), and (0, 3). If you instead use (b) the group of homeomorphisms preserving the boundary pointwise, modulo isotopies preserving the boundary pointwise, then the finite mapping class groups occur for the data (0, 0) and (0, 1). As soon as $g \geq 1$ or $n \geq 4$ there are non-peripherial Dehn twists, and the group is infinite (in both definitions). $\endgroup$
    – Sam Nead
    Commented May 29 at 10:14
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When the genus of the surface is at least 3, Lanier-Margalit proved something much stronger than the fact that there are no finite normal subgroups: aside from the hyperelliptic involution, the normal closure of every finite-order mapping class group is the entire mapping class group. The normal closure of the hyperelliptic involution is the (still infinite) subgroup of the mapping class group mapping to the center in the symplectic group.

See here.

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