A class of highly symmetric Archdeacon embeddings

TL;DR

This paper explores the automorphism groups of Archdeacon embeddings, demonstrating that some embeddings have larger automorphism groups than previously known, including the maximum possible size for infinitely many cases.

Contribution

It shows that Archdeacon embeddings can have automorphism groups larger than rac{v}{ ext{regular}} and constructs examples with maximal automorphism groups using recent array classes.

Findings

Existence of embeddings with automorphism groups larger than rac{v}{ ext{regular}}.

Construction of embeddings with automorphism group size rac{v}{2} for infinitely many v.

Identification of arrays leading to embeddings with maximal automorphism groups.

Abstract

Archdeacon, in his seminal paper $[1]$, defined the concept of Heffter array to provide explicit constructions of biembeddings of the complete graph $K_v$ into orientable surfaces, the so-called Archdeacon embeddings, and proved that these embeddings are $\mathbb{Z}_{v}$-regular. In this paper, we show that an Archdeacon embedding may admit an automorphism group that is strictly larger than $\mathbb{Z}_{v}$. Indeed, as an application of the interesting class of arrays recently introduced by Buratti in $[2]$, we exhibit, for infinitely many values of $v$, an embedding of this type having full automorphism group of size ${v \choose 2}$ that is the largest possible one.