Abstract. The moduli space M g of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of the orbifold is the branch locus B g . In this article we show that B g is connected for genera three, four, thirteen, seventeen, nineteen and fiftynine, and disconnected for any other genus. In order to prove this we use Fuchsian groups, automorphisms of order 5 and 7 of Riemann surfaces, and calculations with GAP for some small genera.

In this work we obtain the group of conformal and anticonformal automorphisms of real cyclic p-gonal Riemann surfaces, where p ≥ 3 is a prime integer and the genus of the surfaces is at least (p − 1) 2 + 1. We use Fuchsian and NEC groups, and cohomology of finite groups.

Abstract. Let g be an integer ≥ 3 and let B g = {X ∈ M g |Aut(X) = 1 d }, where M g denotes the moduli space of compact Riemann surfaces of genus g. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 and 3 belong to the same connected component. We also prove the connectedness of B g for g = 5, 6, 7 and 8 with the exception of the isolated points given by Kulkarni.

Abstract. The moduli space M g of compact Riemann surfaces of genus g has structure of orbifold and the set of singular points for such orbifold is the branch locus B g . In this article we present some results related with the topology of B g , the connectedness of B g for g ≤ 8, the existence of isolated equisymmetric strata of a given dimension in B g and finally the connectedness of the branch locus of the moduli space of Riemann surfaces considered as Klein surfaces.

Abstract. The moduli space M g of compact Riemann surfaces of genus g has structure of orbifold and the set of singular points for such orbifold is the branch locus B g . In this article we present some results related with the topology of B g , the connectedness of B g for g ≤ 8, the existence of isolated equisymmetric strata of a given dimension in B g and finally the connectedness of the branch locus of the moduli space of Riemann surfaces considered as Klein surfaces.