Theory of coverings in the study of Riemann surfaces

Volume 127 / 2012

Ewa Tyszkowska Colloquium Mathematicum 127 (2012), 173-184 MSC: Primary 30F10, 20H10; Secondary 58E40. DOI: 10.4064/cm127-2-3


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.


  • Ewa TyszkowskaInstitute of Mathematics
    University of Gdańsk
    Wita Stwosza 57
    80-952 Gdańsk, Poland

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