Connected Components of Affine Primitive Permutation Groups

Abstract

For a finite group $G$, the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ 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 $\mathcal{H}^{in}_{r,1}(G)$. Furthermore, we give a new algorithm for computing large braid orbits on Nielsen classes. This algorithm utilizes a correspondence between the components of $\mathcal{H}^{in}_{r,1}(G)$ and $\mathcal{H}^{in}_{r,1}(M)$, where $M$ is the point stabilizer in $G$.