Genus zero of projective symplectic groups

Abstract

A transitive subgroup G ≤ SN is called a genus zero group if there exist non identity elements x1 , . . . , xr∈G satisfying G =<x1, . . . , xr>, x1·...·xr=1 and ind x1+...+ind xr = 2N − 2. The Hurwitz space Hinr(G) is the space of genus zero coverings of the Riemann sphere P1 with r branch points and the monodromy group G. In this paper, we assume that G is a finite group with PSp(4, q) ≤ G ≤ Aut(PSp(4, q)) and G acts on the projective points of 3-dimensional projective geometry PG(3, q), q is a prime power. We show that G possesses no genus zero group if q > 5. Furthermore, we study the connectedness of the Hurwitz space Hinr(G) for a given group G and q ≤ 5.