We study that subset of the moduli space of stable genus g, g>1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite group F acts. We show first that this subset is compact. It turns out that, for general finite groups F, the above subset is not connected. We show, however, that for Z_2 actions this subset is connected. Finally, we show that even in the moduli space of smooth genus g Riemann surfaces, the subset of those Riemann surfaces on which Z_2 acts is connected. In view of deliberations of Klein, this was somewhat surprising.
These results are based on new coordinates for moduli spaces. These coordinates are obtained by certain regular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in approximating eigenfunctions of the Laplace operator numerically.
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.