Branched covers of the sphere and the prime-degree conjecture

TL;DR

This paper investigates the realizability of branched covers between surfaces, focusing on the prime-degree conjecture, and provides new evidence supporting it through analysis of hyperbolic orbifolds.

Contribution

It extends previous analysis of candidate surface covers, specifically for hyperbolic and rigid orbifolds with prime degree, offering new realizability results.

Findings

Identified many new realizable candidate covers.

Discovered numerous non-realizable candidate covers.

Provided additional evidence supporting the prime-degree conjecture.

Abstract

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, 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 the candidate covered surface has positive genus, but not all are when it 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 Y -> X between 2-orbifolds, and in a previous paper we have completely analyzed the candidate surface covers such that either X is bad, spherical, or Euclidean, or both X and Y 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 Y has a 2-dimensional Teichmueller space. We find many more realizable and non-realizable candidate covers, providing more support for the prime-degree conjecture.