Maps, simple groups, and arc-transitive graphs

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 Γ admitting an almost simple arc-transitive group X of automorphisms, such that Γ 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.