Archimedean toroidal maps and their minimal almost regular covers

Kostiantyn Drach, Yurii Haidamaka, Mark Mixer, Maksym Skoryk

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 (4.82) 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.


Keywords


Maps, polytopes, groups, covers, Gaussian and Eisenstein integers

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DOI: https://doi.org/10.26493/1855-3974.1825.64c

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications