Finite Group Actions on Compact Riemann Surfaces of Geneus 4

Abstract

Let X be a compact Riemann surface of genus g > 1, let Aut(X) be the group of all biholomorphic mappings of X onto itself and let G be a subgroup of Aut (X). Consider all pairs (X, G). We say that (X ₁, G ₁) is topologically equivalent to (X ₂, G ₂) if there exist an o.p.(orientation preserving) homeomorphism h: X ₁→x ₂ and an isomorphism ι: G ₁→ G ₂ such that ι(σ) o h = h o σ for every σ ∈ G ₁. We write (X ₁, G ₁) ~ (X ₂, G ₂) if (X ₁, G ₁) is topologically equivalent to (X ₂, G ₂). It is known that a pair (X, G) determines a surjective homomorphism φ: Γ→ G with a torsion-free kernel, where Γ is a Fuchsian group of the first kind and having a compact orbit space. In §2, we define an equivalence relation for φ’s such that there exists a bijective correspondence between the topological equivalence classes of (X, G) and the equivalence classes of φ. We classify φ’s up to this equivalence relation in §3; we see that there exist seventy five topological equivalence classes in the case g = 4.