A census of face-transitive surfaces

TL;DR

This paper classifies highly symmetric face-transitive surfaces, identifying seven main types and thirteen sub-types, and provides a census of such surfaces with up to 1280 faces through graph-theoretic constructions.

Contribution

It offers a comprehensive classification of face-transitive surfaces and computes an extensive census using novel graph construction techniques.

Findings

Seven main types of face-transitive surfaces identified

Thirteen sub-types distinguished by automorphism group actions

Census of surfaces with up to 1280 faces constructed

Abstract

A face-transitive surface is a triangulated 2-dimensional manifold whose automorphism group acts transitively on its set of triangles. In this paper, we investigate this class of highly symmetric surface triangulations. We identify seven types of such face-transitive surfaces, splitting up further into a total of thirteen sub-types, distinguished by how their automorphism groups act on them. We use these theoretical results to compute a census of face-transitive surfaces with up to 1280 faces by constructing suitable cycle double covers of cubic node-transitive graphs.