Quasi-vertex-transitive Maps on the Plane

TL;DR

This paper investigates the existence and classification of quasi-vertex-transitive maps on the plane, identifying specific types that exist and determining the minimal genus surface for certain polyhedral maps.

Contribution

It establishes the existence of quasi-vertex-transitive maps of certain types and proves non-existence for vertex-transitive maps of those types, also identifying the minimal genus surface for a specific map type.

Findings

Existence of quasi-vertex-transitive maps of type [p^3, 3] for p ≡ 1 (mod 6)

Non-existence of vertex-transitive maps of these types

Lowest genus surface admitting a [5^3, 3] polyhedral map

Abstract

Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p \equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such types. In particular, we determine the surface with the lowest possible genus that admit a polyhedral map of type $[5^3, 3]$.