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

TL;DR

This paper proves that as the size of the automorphism groups increases, the likelihood of orientably-regular maps and hypermaps being chiral approaches 100%, revealing a generic property of high symmetry complexity.

Contribution

It establishes the asymptotic prevalence of chirality in orientably-regular maps and hypermaps with automorphism groups S_n or A_n, and provides a key probabilistic generation result.

Findings

Proportion of chiral maps tends to 1 as n→∞ for groups S_n and A_n.

Asymptotic probability that random involution and element generate S_n or A_n is 3/4 and 1/4, respectively.

Chirality is a generic property in large automorphism group families.

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.