Vertex-primitive s-arc-transitive digraphs admitting a Suzuki or Ree group

TL;DR

This paper proves that for vertex-primitive s-arc-transitive digraphs admitting Suzuki or Ree groups, the maximum s is 1, and provides constructions to demonstrate this bound is sharp.

Contribution

It establishes that s cannot exceed 1 for these groups, advancing understanding of symmetry properties in such digraphs.

Findings

s  1 for Suzuki and Ree groups

Constructed examples with s=1 to show sharpness

Bound  1 is optimal for these groups

Abstract

The investigation of s-arc-transitivity of digraphs can be dated back to 1989 when the third author showed that s can be arbitrarily large if the action on vertices is imprimitive. However, the situation is completely different when the digraph is vertex-primitive and not a directed cycle. In 2017 the second author, Li and Xia constructed the first infinite family of G-vertex-primitive 2-arc-transitive examples, and asked if there is an upper bound on s for G-vertex-primitive s-arc-transitive digraphs w=that are not directed. In 2018 the second author and Xia showed that if there is a largest such value of s then it will occur when G is almost simple. So far it has been shown that s\leq 2 for almost simple groups whose socle is an alternating group or a projective special linear group. The contribution of this paper is to prove that s\leq 1 in the case of the Suzuki and the small Ree groups. We give constructions with s=1 to show that the bound is sharp.