Problem about orbit size in orbifold

Question

Suppose S is a genus g surface with a hyperbolic manifold structure.

Assume a finite group H acting on S.Does the orbit size of each point in orbifold S/H divides |H|?

Also is it always true that size of each orbit in S/H is precisely |H|?

Thanks in advance.

Answer 1

Some hints:

• Given two points $x, y$ in the same orbit, can you describe some relation between the set of elements of $H$ mapping $x$ to $x$ and the elements mapping $x$ to $y$?

• Does $H$ necessarily act freely?