# Finite normal subgroup of mapping class group

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?

• 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]
• 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).
• @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). Commented May 29 at 10:14