Triangulations and moduli spaces of Riemann surfaces with group actions
Let \(\overline {M^g}\) be the moduli space of stable compact genus \(g\) Riemann surfaces \((g \geq 2)\). This space may be thought of as the Baily-Mumford compactification of the moduli space \(M^g\) of compact smooth Riemann surfaces of genus \(g\). The authors give a new approach to this compactification which allows a generalization to the case of Riemann surfaces with group actions. First they explain their new approach by giving a new proof of the Mahler compactness theorem. The main result of the work under review is Theorem 2: For any finite group \(F\), the set \(\overline {M^g_F} : = \{[X] \in \overline {M^g} : F\) is isomorphic to a subgroup of \(\text{Isom} (X)\}\) is a compact subset of \(\overline M^g\). -- However, for general finite groups \(F\) the sets \(\overline {M^g_F}\) need not be connected. However, for \(\mathbb{Z}_2\)-actions, the above subset is connected. These results are based on new coordinates for moduli spaces which are obtained by certain regular triangulations of Riemann surfaces. The authors announce that these triangulations play also an important role in approximating eigenfunctions of the Laplacian.
scientific article; zbMATH DE number 866163