$\DeclareMathOperator\MCG{MCG}$Let $\Sigma$ be a compact oriented surface, with empty or connected boundary. Let $\mathcal{O}$ the space of orbits of nontrivial simple closed curves on $\Sigma$ under $\MCG(\Sigma)$-action. (so, $\mathcal{O}$ has finitely many elements: the sets of nonseparating curves and the sets of separating curves of each possible genus)


If $f\in \MCG(\Sigma)$ is a noncentral element, is it true that for each $o\in \mathcal{O}$, there is a curve $c\in o$ such that the geometric intersection number $i(f(c),c)$ is non zero ?


Also, in the case where $\Sigma$ has one boundary component, I would like to ask the same question but for arcs (for the geometric intersection number, consider homotopy of arcs fixing the boundary and count only intersection points in the interior)



Action of noncentral mapping classes on curves or arcs on a surface
Yes, this is true - there are many ways to prove it, but I'll hit it with a hammer. Let $C(\Sigma)$ denote the [curve complex](https://en.wikipedia.org/wiki/Curve_complex) of $\Sigma$. Suppose not, then $f$ would map every curve $[c]$ in $\mathcal{O}\subset C(\Sigma)$ to $[f(c)]$ which has distance $\leq 1$ from $[c]$ since $i(c,f(c))=0$. Since the neighborhood of radius 1 of $\mathcal{O}$ is equal to $C(\Sigma)$, we see that $f$ is [quasi-isometric](https://en.wikipedia.org/wiki/Quasi-isometry) to the identity acting on $C(\Sigma)$. Schleimer and Rafi [showed that $Aut(C(\Sigma))\cong QI(C(\Sigma))$](https://arxiv.org/abs/0710.3794), hence $f$ acts trivially on $C(\Sigma)$. Then by [Ivanov's theorem](http://www.math.stonybrook.edu/%7Emlyubich/Archive/Geometry/Hyperbolic%20Geometry/Ivanov.pdf) $f$ is the identity (or central if $\chi(\Sigma)\geq -2$).


