$\DeclareMathOperator\Mod{Mod}$Let $S$ be a surface and $P=\{a\_1,...,a\_n\}$ be a pants decomposition of $S$. Denote by $\Mod(S)$ the mapping class group of $S$.
Define the stabilizer of $\Mod(S)$ on $P$ to be $$A=\{f\in \Mod(S), f(a\_i)=a\_i, i =1,...,n\}.$$
What is $A$?
I was once told that it is generated by Dehn twists and hyperelliptic involution, but I cannot find a reference for that. Is it true, and why? Could you give a reference for this?


I saw this on Lemma 3.2 [in this paper by U. Wolf](https://www.math.kit.edu/iag3/%7Ewolf/media/wolf-the-action-of-the-mapping-class-group-on-the-pants-complex.pdf), which says that $A$ is generated by Dehn twists and half twists But I didn't understand it yet.



Stabilizer of the action of the mapping class group on a pants decomposition
Suppose that $S$ is closed (without boundary), connected, and oriented.


If $S$ has genus two then the stabiliser is generated by Dehn twists about the $a\_i$, the hyperelliptic, and a reflection.


If $S$ has genus greater than two, then there is no hyperelliptic symmetry, but there will still be a reflection symmetry.


This is because any mapping class that stabilises the "cuffs" of the pants (the curves $a\_i$) either permutes the pants (and so there are exactly two pants) or preserves each pants setwise (and thus is isotopic to the identity, or a reflection, off of a small neighbourhood of the cuffs).


