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.

Theorems & Definitions (38)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof