The irreducibility of certain pure-cycle Hurwitz spaces

We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-$0$ covers, using a combination of limit linear series theory and group theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a conditional result in the higher-genus case, requiring at least $3g$ simply branched points. These results have equivalent formulations in group theory, and in this setting complement results of Conway-Fried-Parker-V\"olklein.