Semi-equivelar maps on the torus are Archimedean

TL;DR

This paper proves that all semi-equivelar maps on the torus are quotients of Archimedean tilings on the plane, classifies their symmetry properties, and establishes sharp bounds on vertex orbits under automorphisms.

Contribution

It demonstrates that semi-equivelar maps on the torus originate from Archimedean tilings and analyzes their symmetry and orbit structures.

Findings

All semi-equivelar torus maps are quotients of Archimedean tilings.

Four types of semi-equivelar maps are always vertex-transitive.

Number of vertex orbits is at most six, often at most three.

Abstract

If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map $X$ on the torus is a quotient of an Archimedean tiling on the plane then the map $X$ is semi-equivelar. We show that each semi-equivelar map on the torus is a quotient of an Archimedean tiling on the plane. Vertex-transitive maps are semi-equivelar maps. We know that four types of semi-equivelar maps on the torus are always vertex-transitive and there are examples of other seven types of semi-equivelar maps which are not vertex-transitive. We show that the number of ${\rm Aut}(Y)$-orbits of vertices for any semi-equivelar map $Y$ on the torus is at most six. In fact, the number of orbits is at most three except one type of semi-equivelar maps. Our bounds on the number of orbits are sharp.