Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.08129 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918379904827392 |
|---|---|
| author | Chen, Jiyong Tang, Yi Xiao |
| author_facet | Chen, Jiyong Tang, Yi Xiao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08129 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proportion of chiral maps with automorphism group $\mathcal{S}_n$ and $\mathcal{A}_n$ Chen, Jiyong Tang, Yi Xiao Group Theory Combinatorics 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. |
| title | Proportion of chiral maps with automorphism group $\mathcal{S}_n$ and $\mathcal{A}_n$ |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2603.08129 |