Proportion of chiral maps with automorphism group ${\mathcal{S}}_n$ and ${\mathcal{A}}_n$

Abstract

Orientably-regular maps are highly symmetric embeddings of graphs in oriented surfaces. Among them, chiral maps are those which fail to be isomorphic to their mirror images. We prove that, as $n\to \infty$, chirality is generic for orientably-regular maps with automorphism groups $S_n$ or $A_n$: the proportion of chiral maps tends to $1$ in both families. We also obtain the corresponding asymptotic result for orientably-regular hypermaps with automorphism groups $S_n$ or $A_n$. A key ingredient is a sharp asymptotic generation statement: if one chooses an involution of $S_n$ uniformly at random and then chooses an independent uniformly random element of $S_n$, the probability that these two elements generate $S_n$ and $A_n$ tends to $\frac{3}{4}$ and $\frac{1}{4}$ as $n\to \infty$, respectively.