Branched covers of the sphere and the prime-degree conjecture

Abstract

To a branched cover $${\widetilde{\Sigma} \to \Sigma}$$ between closed, connected, and orientable surfaces, one associates a branch datum, which consists of Σ and $${\widetilde{\Sigma}}$$ , the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann–Hurwitz formula. A candidate surface cover is an abstract branch datum, a priori not coming from a branched cover, but satisfying the Riemann– Hurwitz formula. The old Hurwitz problem asks which candidate surface covers are realizable by branched covers. It is now known that all candidate covers are realizable when Σ has positive genus, but not all are when Σ is the 2-sphere. However, a long-standing conjecture asserts that candidate covers with prime degree are realizable. To a candidate surface cover, one can associate one $${\widetilde {X} \dashrightarrow X}$$ between 2-orbifolds, and in Pascali and Petronio (Trans Am Math Soc 361:5885–5920, 2009), we have completely analyzed the candidate surface covers such that either X is bad, spherical, or Euclidean, or both X and $${\widetilde{X}}$$ are rigid hyperbolic orbifolds, thus also providing strong supporting evidence for the prime-degree conjecture. In this paper, using a variety of different techniques, we continue this analysis, carrying it out completely for the case where X is hyperbolic and rigid and $${\widetilde{X}}$$ has a 2-dimensional Teichmüller space. We find many more realizable and non-realizable candidate covers, providing more support for the prime-degree conjecture.