Introducing edge-biregular maps

TL;DR

This paper introduces the concept of edge-biregular maps, explores their symmetry properties, and classifies such maps on surfaces with non-negative Euler characteristic and on arbitrary surfaces with dihedral automorphism groups.

Contribution

It defines edge-biregular maps, studies their automorphism groups, and provides classifications for maps on various surfaces, linking them to regular maps and triangle groups.

Findings

Classified edge-biregular maps on surfaces with non-negative Euler characteristic.

Connected edge-biregular maps to regular maps via subgroup constructions.

Described automorphism groups generated by four involutions.

Abstract

We introduce the concept of alternate-edge-colourings for maps, and study highly symmetric examples of such maps. Edge-biregular maps of type $(k,l)$ occur as smooth normal quotients of a particular index two subgroup of $T_{k,l}$, the full triangle group describing regular plane $(k,l)$-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.