Action of Mapping Class Group on Arc complex 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.
 A: The quotient complex was studied in a paper by Penner, "The structure and singularities of quotient arc complexes", Journal of Topology 1 (2008), 527-550. An earlier version of the paper is available on the arXiv.  He enumerated all cases when the quotient is a sphere (this only happens when the genus and number of boundary components are small) and he found there are exactly four cases when the quotient is a manifold other than a sphere: 
(1) annulus with two punctures. 
(2) pair of pants with one puncture.
(3) genus 0 with 4 boundary circles and no punctures.
(4) genus 1 with 2 boundary circles and no punctures.
In each case one is considering arcs with endpoints either at punctures or at one distinguished point in each boundary circle. As a fun exercise I once tried to figure out what the manifold was in these four cases.  According to some very sketchy notes I still have, I got answers in the first three cases: 
(1) $S^2\times S^3$.
(2) twisted product of $S^2$ and $S^5$.
(3) $S^2\times S^7$.
In case (4) Penner showed the quotient is a closed 7-manifold but I didn't determine what it was. Someone should figure this out and also check the answers in (1-3).  As I recall there were some rather pretty pictures that came up.
A: Here is a closely related mathoverflow question.
How many triangulations of the genus $g$ surface on $n$ vertices?
Suppose that $S$ has genus $g$ and $n$ boundary components. Then all maximal simplices in the arc complex $A(S)$ have the same dimension, and correspond to hexagonal decompositions of $S$.  Alternatively, if you crush the boundary components to points, they correspond to triangulations of the surface with marked points.  Two maximal simplices are adjacent if and only if the corresponding triangulations differ by a diagonal flip.  This gives a way to talk about the top dimensional simplices. 
As $g$ and $n$ increase, this "graph of triangulations" has very rich combinatorics and is a subject of current research.  As just a hint of the difficulties involved, it is very difficult to even count the number of vertices of the graph of triangulations (the number of top dimensional simplices of $A(S)$).  Even getting good asymptotics is a challenge, as the above linked mathoverflow question indicates.
