Genus g Groups of Diagonal Type

A transitive subgroup G\leq S_n is called a genus g group if there exist non identity elements x_۱,...,x_r\in G satisfying G=\langle x_۱,x_۲,...,x_r\rangle, \prod_{i=۱}^r {x_i}=۱ and \sum_{i=۱}^r ind\, x_i=۲(n+g-۱). The Hurwitz space \mathcal{H}^{in}_{r,g}(G) is the space of genus g covers of the Riemann sphere \mathbb{P}^۱\mathbb{C} with r branch points and the monodromy group G. Isomorphisms of such covers are in one to one correspondence with genus g groups.In this article, we show that G possesses genus one and two group if it is diagonal type and acts primitively on \Omega. Furthermore, we study the connectedness of the Hurwitz space \mathcal{H}^{in}_{r,g}(G) for genus ۱ and ۲.

Keywords:

Braid Orbit , Genus g System , Primitive Group