On the orbifold structure of the moduli space of Riemann surfaces of genera four and five
For a given genus \(g \geq 4\), the moduli space \(\mathcal M_g\) of compact Riemann surfaces of genus \(g\) is the quotient of the Teichmüller space by the action of the mapping class group. It has the structure of a complex orbifold, and its set of singular points is called the branch locus \(\mathcal B_g\). When the actions of the automorphism groups of two surfaces of the same genus are topologically equivalent, the surfaces are said to be equisymmetric. Then the moduli space \(\mathcal M_g\) has an equisymmetric stratification in which each stratum consists of the points corresponding to equisymmetric surfaces. So the branch locus is formed by the strata corresponding to surfaces whose automorphism group is not trivial. The goal of the paper under review is to determine the orbifold structure of the moduli spaces \(\mathcal{M}_4\) and \(\mathcal{M}_5\), by means of the topological classification of the actions of the automorphism groups of Riemann surfaces of both genera \(4\) and \(5\). These actions are determined using the uniformization of Riemann surfaces by Fuchsian groups. The strata of \(\mathcal{M}_4\) and \(\mathcal{M}_5\) correspond to the actions of the full automorphism groups of Riemann surfaces of the respective genus. So the authors obtain firstly all triples \((G,\Delta,\theta)\) satisfying that \(\theta : \Delta \rightarrow G\) is an epimorphism from the Fuchsian group \(\Delta\) onto the finite group \(G\) such that \(\ker(\theta)\) is a surface group of genus \(4\) (resp. \(5\)). Then they determine when these actions are maximal, and so the group \(G\) is the full group of automorphisms of a surface, what gives the list of equisymmetric strata of \(\mathcal{B}_4\) and \(\mathcal{B}_5\). The last step is to determine the inclusion relations of the strata for each of the two branch loci \(\mathcal{B}_4\) and \(\mathcal{B}_5\). The key point to obtain it is the fact that if the action \((G, \theta)\) is an extension of the action \((G', \theta')\), then the stratum of the former is included in the stratum of the latter. Thus the problem is solved in terms of extensions of automorphism groups.
scientific article; zbMATH DE number 6341672
scientific article