Maps, simple groups, and arc-transitive graphs

TL;DR

This paper classifies certain factorisations of finite almost simple groups and applies these results to classify specific arc-transitive graphs with surface embeddings, identifying known and new graph families.

Contribution

It provides a complete classification of group factorisations with cyclic or dihedral intersections and applies this to classify arc-transitive graphs with surface embeddings, including new infinite families.

Findings

Classified all factorizations of almost simple groups with core-free subgroups intersecting cyclically or dihedrally.

Identified all arc-transitive graphs with surface embeddings under certain group actions, including known and new graphs.

Constructed infinitely many embeddings in the exceptional case where the socles are alternating groups.

Abstract

We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple arc-transitive group $X$ of automorphisms, such that $\Gamma$ has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup $G$ of $X$. We prove that apart from the case where $X$ and $G$ have socles $A_n$ and $A_{n-1}$ respectively, the only such graphs are the complete graphs $K_n$ with $n$ a prime power, the Johnson graphs $J(n,2)$ with $n-1$ a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings.