Criterion for existence of G‑equivariant maps on arbitrary Riemann surfaces

Develop a criterion that determines, for a given compact Riemann surface E and subgroup G ≤ Aut(E), whether there exists a non-constant holomorphic G‑equivariant map V from E to some compact Riemann surface S, i.e., a map for which there is an injective homomorphism φ: G → Aut(S) satisfying V ∘ μ = φ(μ) ∘ V for all μ in G.

Background

In the sphere case (E = CP^1), G‑equivariant rational functions exist for any finite G and admit explicit descriptions. The authors extend the framework to holomorphic maps between compact Riemann surfaces and define G‑equivariance via an injective homomorphism into the automorphism group of the target.

Beyond CP^1, however, a general existence criterion for G‑equivariant maps is unknown, and the paper provides examples and partial constructions (e.g., for G ≅ Z/2) but no full characterization.