Summary
We study that subset of the moduli space\(\bar M^g \) of stable genusg,g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite groupF acts. We show first that this subset is compact. It turns out that, for general finite groupsF, the above subset is not connected. We show, however, that for ℤ2 actions this subsetis connected. Finally, we show that even in the moduli space ofsmooth genusg Riemann surfaces, the subset of those Riemann surfaces on which ℤ2 actsis connected. In view of deliberations of Klein ([8]), this was somewhat surprising.
These results are based on new coordinates for moduli spaces. These coordinates are obtained by certainregular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically.
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This work has been supported by the European Communities Science Plan Project No SCI*-CT91 (TSTS) “Computational Methods in the Theory of Riemann Surfaces and Algebraic Curves,” by Academy of Finland and by the Swiss National Science Foundation Grant 20-34099.92. We thank M. C. Petrus for providing excellent motivation for this work.
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Buser, P., Seppälä, M. & Silhol, R. Triangulations and moduli spaces of Riemann surfaces with group actions. Manuscripta Math 88, 209–224 (1995). https://doi.org/10.1007/BF02567818
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DOI: https://doi.org/10.1007/BF02567818