Exceptional groups and the $s$-arc-transitivity of vertex-primitive digraphs, I

Abstract

In this paper, we study the primitive actions of almost simple exceptional groups of Lie type on \(s\)-arc-transitive digraphs. Our motivation is the following question posed by Giudici and Xia: Is there an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs that are not directed cycles? In a 2018 paper, Giudici and Xia reduced this question to the case where the automorphism group of the digraph is an almost simple group with socle \(L\). Subsequently, it has been proved that $s\le 2$ when \(L\) is a linear, symplectic or alternating group, and $s\le 1$ when \(L\) is a Suzuki group, a small Ree group, or one of $22$ specific sporadic groups. In this paper, we prove that $s\le 2$ when \(L\) is ${}^3{D}_4(q)$, $G_2(q)$ (including $G_2(2{)}^{\prime }$), ${}^2{F}_4(q)$ (including ${}^2{F}_4(2{)}^{\prime }$), $F_4(q)$, $E_6(q)$ or ${}^2{E}_6(q)$.