Vertex-transitive covers of semi-equivelar toroidal maps

TL;DR

This paper proves that semi-equivelar toroidal maps can be covered by vertex-transitive maps, establishing a connection between these classes and showing that semi-equivelar maps are quotients of almost regular maps.

Contribution

The paper demonstrates that every semi-equivelar toroidal map has a finite vertex-transitive cover, linking semi-equivelar maps to vertex-transitive and almost regular toroidal maps.

Findings

Semi-equivelar toroidal maps are quotients of vertex-transitive maps.

Each semi-equivelar map has a finite vertex-transitive cover.

Semi-equivelar maps are quotients of almost regular toroidal maps.

Abstract

A map $X$ on a surface is called vertex-transitive if the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar toroidal maps which are not vertex-transitive. In this article, we show that semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps. More explicitly, we prove that each semi-equivelar toroidal map has a finite vertex-transitive cover. In 2019, Drach {\em et al.} have shown that each vertex-transitive toroidal map has a minimal almost regular cover. Therefore, semi-equivelar toroidal maps are quotients of almost regular toroidal maps.