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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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