REAL GENUS ACTIONS OF
FINITE SIMPLE GROUPS

COY L. MAY

Abstract:

A finite group $G$ can be represented as a group of dianalytic automorphisms of a compact bordered Klein surface, that is, $G$ acts effectively on a bordered surface. The real genus $\rho\,(G)$ is the minimum algebraic genus of any bordered surface on which $G$ acts. A real genus action of $G$ is an action of $G$ on a bordered surface of (algebraic) genus $\rho\,(G)$. In this paper we consider real genus actions of finite simple groups. Let $G$ be a finite simple group, and let $X$ be a bordered surface of least genus on which $G$ acts. We show that if $G$ is $(2,s,t)$-generated, then $G$ is normal in ${\rm Aut}\,(X)$, $[{\rm Aut}\,X:G]$ divides 4, and ${\rm Aut}\,X$ embeds faithfully in ${\rm Aut}\,G$. We also consider the real genus actions of each projective special linear group ${\rm PSL}\,(2,q)$.