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

Abstract

In this paper, we study the primitive actions of almost simple groups with socle \(E_7(q)\) or \(E_8(q)\) on an \(s\)-arc-transitive digraph. Our motivation goes back to the question of whether \(s\) is bounded above for finite connected \(G\)-vertex-primitive \(s\)-arc-transitive digraphs that are not directed cycles. The question has been reduced by Giudici and Xia to the case where \(G\) is almost simple. This work succeeds our 2025 paper (Yin and Chen), which addressed \({}^3\!D_4(q)\), \(G_2(q)\), \({}^2\!F_4(q)'\), \(F_4(q)\), \(E_6(q)\), and \({}^2\!E_6(q)\). Together with Chen, Giudici, and Praeger's work on \({}^2\!B_2(q)\) and \({}^2\!G_2(q)\), it answers the question for all exceptional groups.