Quasiprimitive and bi-quasiprimitive highly-arc-transitive digraphs and finite simple groups

TL;DR

This paper explores the structure of highly-arc-transitive digraphs, linking their properties to finite simple groups and introducing new constructions for bipartite and vertex-bi-quasiprimitive digraphs with high symmetry.

Contribution

It extends quotient digraph concepts, establishes connections between arc-transitivity and finite simple groups, and introduces a novel construction method for highly symmetric bipartite digraphs.

Findings

Infinite families of $(H,s)$-arc-transitive digraphs with alternating groups.

A new construction producing bipartite $G$-arc-transitive digraphs from non-bipartite ones.

Identification of open problems in highly symmetric digraph structures.

Abstract

We extend the notion of an $H$-normal quotient digraph of an $H$-vertex-transitive digraph to that of an $H$-subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each finite connected $H$-vertex-transitive, $(H,s)$-arc-transitive digraph with $s\geqslant6$, either some $H$-normal quotient is a directed cycle of length at least $3$, or there is an $(L,t)$-arc-transitive digraph with $t\geqslant (s-3)/2$, and $L$ a vertex-quasiprimitive almost simple group with socle a composition factor of $H$. This connection demonstrates that, to understand finite $s$-arc-transitive digraphs with large $s$, those admitting a vertex-quasiprimitive almost simple $s$-arc-transitive subgroup of automorphisms play a central role. We show that for each $s$ and each odd valency $k$, there are infinitely many $(H,s)$-arc-transitive digraphs of valency $k$ with $H$ a finite alternating group. In addition we discovered a novel construction which takes as input a connected non-bipartite $H$-vertex-transitive, $(H,s)$-arc-transitive digraph, and outputs a connected bipartite $G$-vertex-transitive, $(G,2s)$-arc-transitive digraph with $G=(H\times H).2$. This leads to construction of vertex-bi-quasiprimitive $s$-arc-transitive digraphs, for arbitrarily large $s$. Our investigations yield several new open problems.