Genus One Almost Simple Groups of Lie Rank Two

Abstract

In this paper, we assume that $G$ is a finite group with socle $PSL(3,q)$ and $G$ acts on the projective points of 2-dimensional projective geometry $PG(2,q)$, $q$ is a prime power. By using a new method, we show that $G$ possesses no genus one group if $q>13$. Furthermore, we study the connectedness of the Hurwitz space $\mathcal{H}^{in}_{r}(G)$ for a given group $G$, genus one and $q\leq 13$.