Theory of coverings in the study of Riemann surfaces.(English)Zbl 1250.30036

Summary: 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.

MSC:

 30F10 Compact Riemann surfaces and uniformization 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 58E40 Variational aspects of group actions in infinite-dimensional spaces
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