On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus

Abstract

Let g g be an integer ≥ 3 \ge 3 and let B g = { X ∈ M g | A u t ( X ) ≠ 1 d } \mathcal {B}_g = \{X\in \mathcal {M}_g | Aut(X)\neq 1_d \} , where M g \mathcal {M}_g denotes the moduli space of compact Riemann surfaces of genus g 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 \mathcal {B}_g for g = 5 , 6 , 7 g=5,6,7 and 8 8 with the exception of the isolated points given by Kulkarni.