Manuscripta Mathematica Manuscript-nr. Triangulations and Moduli Spaces of Riemann Surfaces with Group Actions ?

We study that subset of the moduli space M g of stable genus g, g > 1, Riemann surfaces which consists of such stable Riemann surfaces on which a given nite group F acts. We show rst that this subset is compact. It turns out that, for general nite 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 ((8]), 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.