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 finite ordered element $f$ of $MCG(S)$ such that $f(\alpha)=\alpha$ and $f(\beta)=\beta$ in $\mathcal {C}(S)$?

Thanks!

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!