Existence of a disjoint collection of separating, nontrivial curves on an orientable surface invariant under a finite action

Question

Suppose that a finite group acts upon an orientable surface of genus greater than one. Is there a disjoint collection of separating, nontrivial, simple closed curves on the surface invariant under the action?

Considering only orientation preserving actions at first, we could pick a nontrivial, separating, simple closed curve that avoids any fixed point of any element of the group. Then the orbit of this curve is a collection of (probably intersecting) separating, nontrivial closed curves.

You could then remove each of the intersections one by one (being careful to do the same removal for the orbit of each intersection) to leave a disjoint collection of simple closed curves invariant under the action.

Would these curves necessarily be separating? nontrivial?

Any suggestions would be greatly appreciated.

Answer 1

Not it might not be! In fact, there exists periodic elements of the mapping class groups that are irreducible, in particular, there is no set of disjoint non-trivial simple closed curves that are preserved under $f.$