Past exam of the geometry of surfaces course of the University of Oxford / 2023 / 2 / d / Solution

For a free action of a group, every stabilizer subgroup is trivial. Hence every ramification index of a holomorphic map is one, and the projection is an unramified covering map of degree of a holomorphic map $|G|$. The Riemann-Hurwitz formula reduces toThis is also the multiplicativity of Euler characteristic under a finite covering.