# When is the action of a mapping class group on the set of punctures realized by a finite subgroup of mapping classes?

I have a curious question about a natural sequence, which I haven't seen answered in the literature.

Let $$\Sigma$$ be an oriented surface of genus $$g$$ without boundary with a set $$\mathcal P_n$$ of $$n$$ punctures on it. The mapping class group $$\mathcal M(\Sigma,\mathcal P_n)$$ is the group of orientation-preserving mapping classes which fix $$\mathcal P_n$$ as a set (where mapping classes are considered up to isotopies of $$\Sigma$$ which preserve $$\mathcal P_n$$ pointwise). Similarly, the pure mapping class group $$\mathcal{PM}(\Sigma,\mathcal P_n)$$ is the subgroup of $$\mathcal M(\Sigma,\mathcal P_n)$$ consisting of mapping classes which fix each puncture from $$\mathcal P_n$$. We have a short exact sequence: $$1\to \mathcal{PM}(\Sigma,\mathcal P_n)\to \mathcal M(\Sigma, \mathcal P_n)\to \mathbb S_n\to 1,$$ where $$\mathbb S_n$$ is the symmetric group on $$n$$ letters.

Question: for which values $$(g,n)$$ does this sequence split? I.e. when is there a subgroup of $$\mathcal M(\Sigma,\mathcal P_n)$$ isomorphic to $$\mathbb S_n$$ inducing the same action on punctures?

I suspect that the answer is: $$\text{it splits only for }(g,n)=(g,1), (g,2) \text{ for arbitrary g, and } (g,3) \text{ for }g\not\equiv 2 \bmod 3.$$

The values of $$n\ge4$$ can be excluded by the following consideration: any finite subgroup of the stabilizer of a puncture in $$\mathcal M(\Sigma,\mathcal P_n)$$ must be cyclic (see e.g. Lemma A.3.11 in F.Castel “Geometric representations of the braid groups”, Asterisque, 378, 2016.).

That the values $$n=1,2$$ for arbitrary $$g$$ and $$n=3$$ for $$g\equiv 0,1 \bmod 3$$ give us split sequences can be seen by drawing $$\mathbb S_n$$-symmetrical pictures of the surface $$\Sigma$$. However, I don't know how to prove that for $$g\equiv 2 \bmod 3$$ and $$n=3$$ the sequence is not split. Any ideas?

Update: It turns out, that this sequence splits for $$n=1,2,3$$ regardless of genus, and does not split for $$n\ge4$$. See a beautiful construction in the answer of Dave Benson.

I don't think what you're asking for is true. Namely, I think the sequence does split for $$S_3$$ when $$g\equiv 2$$ mod $$3$$. For example, if $$g=2$$ then you can make a surface that looks like three tubes going from a blob in the top to a blob in the bottom, and put one puncture on each tube. For general $$g\equiv 2$$ mod $$3$$ you'd have $$g+1$$ tubes and choose three of them symmetrically placed, to put the punctures. Even elements of $$S_3$$ are given by rotations about a vertical axis, and odd elements are given by rotations about a horizontal axis going through the fixed puncture.