2-uniform covers of $2$-semiequivelar toroidal maps

TL;DR

This paper investigates conditions under which 2-semiequivelar toroidal maps can be covered by finite 2-uniform maps, extending understanding of tiling symmetries on toroidal surfaces.

Contribution

It proves that most 2-semiequivelar toroidal maps have finite 2-uniform covers, except for two specific types, advancing classification of symmetric tilings on tori.

Findings

Most 2-semiequivelar toroidal maps have finite 2-uniform covers.

Two types of 2-semiequivelar maps do not admit such covers.

The universal cover's uniformity determines the existence of finite covers.

Abstract

If every vertex in a map has one out of two face-cycle types, then the map is said to be $2$-semiequivelar. A 2-uniform tiling is an edge-to-edge tiling of regular polygons having $2$ distinct transitivity classes of vertices. Clearly, a $2$-uniform map is $2$-semiequivelar. The converse of this is not true in general. There are 20 distinct 2-uniform tilings (these are of $14$ different types) on the plane. In this article, we prove that a $2$-semiequivelar toroidal map $K$ has a finite $2$-uniform cover if the universal cover of $K$ is $2$-uniform except of two types.