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 w hich a given finite group F acts. We show first that this subset is co mpact. It turns out that, for general finite groups F, the above subse t is not connected. We show, however, that for Z(2) actions this subse t is connected. Finally, we show that even in the moduli space of smoo th genus g Riemann surfaces, the subset of those Riemann surfaces on w hich Z(2) acts is connected. In view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinat es for moduli spaces. These coordinates are obtained by certain regula r triangulations of Riemann surfaces. These triangulations play an imp ortant role also elsewhere, for instance in approximating eigen functi ons of the Laplace operator numerically.