Vertex-primitive $s$-arc-transitive digraphs of linear groups

TL;DR

This paper investigates the structure of certain highly symmetric directed graphs with almost simple linear group symmetries, establishing that the maximum arc-transitivity level is two, which advances understanding of their symmetry limits.

Contribution

It proves that for vertex-primitive digraphs with socle PSL_n(q), the maximum s for s-arc-transitivity is two, providing a key step toward a general bound.

Findings

s ≤ 2 for such digraphs

First step in bounding s for all vertex-primitive s-arc-transitive digraphs

Enhances understanding of symmetry properties in linear group actions

Abstract

We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that $s\leqslant2$ for such digraphs, which provides the first step in determining an upper bound on $s$ for all the vertex-primitive $s$-arc-transitive digraphs.