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!



The action of torsion of $MCG(S)$ on curve complex
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. 


