Connected components of affine primitive permutation groups

Abstract

For a finite group G, the Hurwitz space Hr,gin(G) is the space of genus g covers of the Riemann sphere with r branch points and the monodromy group G.In this paper, we give a complete list of primitive genus one systems of affine type. That is, we assume that G is a primitive group of affine type. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in Hr,1in(G). Furthermore, we give a new algorithm for computing large braid orbits on Nielsen classes. This algorithm utilizes a correspondence between the components of Hr,1in(G) and Hr,1in(M), where M is the point stabilizer in G.