Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 1 / c

Because the action is free, the quotient $S_g/G$ is a closed orientable surface $S_h$, and $S_g\to S_h$ is a covering of degree $|G|$. The Euler characteristic under a finite covering givesor equivalentlyThe quotient inherits a hyperbolic metric, so $h\ge 2$. Hence $h-1\ge 1$ and $|G|\le g-1$.