# Stabilizer of the action of the mapping class group on a pants decomposition

$$\DeclareMathOperator\Mod{Mod}$$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, which says that $$A$$ is generated by Dehn twists and half twists But I didn't understand it yet.

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).