Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of **Mod(S)**(mapping class group) on the set of all vertices has finitely many orbits (see this post). My question is

**What are the orbits of higher cells? Are there finitely many orbits? In other words how the quotient looks like?**

From the above post we know it will consists of finitely many vertices. I think a variation of the above proof is also gives the answer for 1-cells i.e. edges. But what about cells of higher dimension.

PS: Any link , reference, paper will be extremely helpful. Thanks in advance.