On the existence of covers of ${\mathbb{P}}_1$ associated to certain permutations

Abstract

In this short note we prove the impossibility of realizing finite topological covers of the Riemann sphere minus three points, associated to certain explicit combinatorial (permutation) data. This comes from a question of M. Zieve and falls in the framework of the so-called ‘‘Hurwitz problem’’, asking for a ‘‘simple’’ description of the combinatorial data which can be so realized.