Theory of coverings in the study of Riemann surfaces
For a $G$-covering $Y\rightarrow Y/G=X$ induced by a properly discontinuous action of a group $G$ on a topological space $Y$, there is a natural action of $\pi (X,x)$ on the set $F$ of points in $Y$ with nontrivial stabilizers in $G$. We study the covering of $X$ obtained from the universal covering of $X$ and the left action of $\pi (X,x)$ on $F$. We find a formula for the number of fixed points of an element $g\in G$ which is a generalization of Macbeath's formula applied to an automorphism of a Riemann surface. We give a new method for determining subgroups of a given Fuchsian group.