Irreducibility of moduli spaces of cyclic unramified covers of genus $g$ curves

Let (C1 , . . ., Cr , G) = (C, G) be an r-tuple consisting of a transitive subgroup G of Sm and r conjugacy classes C1, . . ., Cr of G. We consider the concept of the moduli space W(C, G) of compact Riemann surface covers of the Riemann sphere of Nielsen class (C, G). The irreducibility of }((C, G) is equivalent to the transitivity of a specific permutation representation of the Hurwitz monodromy group (§1), but there are few general tools to decide questions about this representation. Theorem 2 gives a class of examples of (C, G) for which }((C, G) is irreducible. As an immediate corollary this gives an elementary proof and generalization of the irreduciblity of the moduli space of cyclic unramified covers of genus g curves (for which Deligne and Mumford [DM, Theorem 5.15] applied Teichmuller theory and Dehn's theorem). This contrasts with the examples of (C, G) in [BFr] for which }((C, G) is reducible. These kinds of questions combined with the study of the existence of rational subvarieties of W(C, G) have application to the realization of a group G as the Galois group of a regular extension of 12(t) [Fr3, §4j.