Archimedean toroidal maps and their minimal almost regular covers

TL;DR

This paper investigates Archimedean maps on the torus, proving the existence and uniqueness of minimal almost regular covers for each type, and explicitly constructing these covers using algebraic integers.

Contribution

It introduces the concept of minimal almost regular covers for Archimedean toroidal maps and provides explicit constructions using Gaussian and Eisenstein integers.

Findings

Each Archimedean toroidal map has a unique minimal almost regular cover.

Explicit constructions of these covers are provided using algebraic integers.

The results extend previous work on equivelar maps on the torus.

Abstract

The automorphism group of a map acts naturally on its flags (triples of incident vertices, edges, and faces). An Archimedean map on the torus is called almost regular if it has as few flag orbits as possible for its type; for example, a map of type $(4.8^2)$ is called almost regular if it has exactly three flag orbits. Given a map of a certain type, we will consider other more symmetric maps that cover it. In this paper, we prove that each Archimedean toroidal map has a unique minimal almost regular cover. By using the Gaussian and Eisenstein integers, along with previous results regarding equivelar maps on the torus, we construct these minimal almost regular covers explicitly.