Archimedean toroidal maps and their minimal almost regular covers
Group Theory
2018-10-03 v1 Combinatorics
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 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.
Cite
@article{arxiv.1810.01386,
title = {Archimedean toroidal maps and their minimal almost regular covers},
author = {Kostiantyn Drach and Yurii Haidamaka and Mark Mixer and Maksym Skoryk},
journal= {arXiv preprint arXiv:1810.01386},
year = {2018}
}