Equivelar toroids with few flag-orbits

TL;DR

This paper classifies equivelar toroids, which are higher-dimensional generalizations of torus maps, focusing on those with at most n flag-orbits, including a detailed classification of 2-orbit toroids across dimensions.

Contribution

It provides a comprehensive classification of equivelar (n+1)-toroids with limited flag-orbits, especially detailing 2-orbit toroids in arbitrary dimensions.

Findings

Classified equivelar (n+1)-toroids with up to n flag-orbits.

Established a framework for understanding 2-orbit toroids in any dimension.

Extended the theory of regular tessellations to higher-dimensional toroids.

Abstract

An $(n+1)$-toroid is a quotient of a tessellation of the $n$-dimensional Euclidean space with a lattice group. Toroids are generalizations of maps in the torus on higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar $(n+1)$-toroids with at most $n$ flag-orbits; in particular, we discuss a classification of $2$-orbit toroids of arbitrary dimension.