Branched covers of the sphere and the prime-degree conjecture

Pascali M A, Petronio C

Hyperbolic 2-orbifold  Geometry 

To a branched cover ${widetilde{Sigma} to Sigma}$ between closed, connected, and orientable surfaces, one associates a branch datum, which consists of ? 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 ? has positive genus, but not all are when ? 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.

Source: ANNALI DI MATEMATICA PURA ED APPLICATA, vol. 191 (issue 3), pp. 563-594