The action of torsion of $MCG(S)$ on curve complex Hi everyone. 
Let $S$ be a closed surface with genus at least 3, $\alpha, \beta$ be the two vertices of 
curve complex of $S$ such that $d_{\mathcal {C}(S)}(\alpha, \beta)\geq 3$. 
My question is 
Is there a non-trivial  finite ordered element $f$ of $MCG(S)$ such that $f(\alpha)=\alpha$ and $f(\beta)=\beta$ in $\mathcal {C}(S)$?
Thanks!
 A: Here is a way to find lots of examples.  Suppose that $\Sigma$ is a surface and suppose that $f$ is a periodic mapping class.  Let $S$ be the quotient orbifold $\Sigma/f$.  Then taking full preimages gives a quasi-isometric embedding of the curve complex of $S$ into the curve complex of $\Sigma$.  See 
arXiv:1104.3492 and arXiv:math/0701719
for two different proofs. When $S$ has an infinite diameter curve complex we can pick $a$ and $b$, curves in $S$, that are as well separated as you want.  Then the preimages $\alpha$ and $\beta$ are also well separated, and fixed as simplices by $f$.  A bit more work (which I haven't done) should give examples where $\alpha$ and $\beta$ are single vertices.  There are also examples where $S$ is honestly a surface, not just an orbifold. 
A: Here is an example which can be easily generalized for all $S_g$, $g\geq 3$. With similar approaches, one can construct many examples, not just starting with the building block $S_{1,2}$. Notice the Dehn twist about one boundary component in $S_{1,2}$.
http://s11.postimage.org/ajzyt2xsz/finiteorderfill.png
Given a finite group $G$ of mapping classes, can one find $S_g$, and curves $a,b$ on $S_g$ that fill such that $a,b$ are stabilized by each element of $G$? (off the top of my head, the example I have given does each finite cyclic group of order at least 3)
