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A census of face-transitive surfaces

Reymond Akpanya, Jonathan Spreer

TL;DR

This work classifies and enumerates face-transitive triangulations of surfaces up to $1280$ faces by exploiting a correspondence between face graphs and cycle double covers on cubic node-transitive graphs. Central to the method is the vertex–face type invariant, which partitions surfaces into seven types (giving thirteen subtypes) and guides construction via finite subgroups $H\le Aut(\mathcal{G})$ acting on the face graph. The authors achieve a comprehensive census of $86{,}802$ isomorphism classes, split into $80{,}243$ orientable and $6{,}559$ non-orientable cases, and provide explicit constructions, examples, and a public dataset. The results illuminate the landscape of highly symmetric triangulations, show several nontrivial constraints on large automorphism groups, and establish a foundation for systematic exploration of other symmetric surface classes.

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.

A census of face-transitive surfaces

TL;DR

This work classifies and enumerates face-transitive triangulations of surfaces up to faces by exploiting a correspondence between face graphs and cycle double covers on cubic node-transitive graphs. Central to the method is the vertex–face type invariant, which partitions surfaces into seven types (giving thirteen subtypes) and guides construction via finite subgroups acting on the face graph. The authors achieve a comprehensive census of isomorphism classes, split into orientable and non-orientable cases, and provide explicit constructions, examples, and a public dataset. The results illuminate the landscape of highly symmetric triangulations, show several nontrivial constraints on large automorphism groups, and establish a foundation for systematic exploration of other symmetric surface classes.

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.
Paper Structure (39 sections, 25 theorems, 43 equations, 17 figures, 2 tables)

This paper contains 39 sections, 25 theorems, 43 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related work and background.
  3. Summary of contributions.
  4. Structure of the Paper
  5. Preliminaries
  6. Simplicial surfaces
  7. Colourings of simplicial surfaces
  8. Face graphs of simplicial surfaces
  9. Group actions on simplicial surfaces and their face graphs
  10. Face-transitive surfaces
  11. Construction of face-transitive surfaces
  12. Face-transitive surfaces with vertex-face type (3,1)
  13. Face-transitive surfaces with vertex-face type (2,2)
  14. Face-transitive surfaces with vertex-face type (1,6)
  15. Face-transitive surfaces with vertex-face type (1,2)
  16. ...and 24 more sections

Key Result

Theorem 1.1

There are exactly $86\, 802$ isomorphism classes of face-transitive surfaces with up to $1280$ faces.

Figures (17)

  • Figure 1: (a) The simplicial tetrahedron $\mathop{\mathrm{\mathcal{T}}}\nolimits$ and (b) its face graph $\mathop{\mathrm{\mathcal{F}}}\nolimits(X)$
  • Figure 2: (a) Grünbaum-colouring and (b) vertex-$4$-colouring of the simplicial tetrahedron $\mathop{\mathrm{\mathcal{T}}}\nolimits$
  • Figure 3: (a) Rotational edge and (b) mirror edge of a Grünbaum-coloured simplicial surface
  • Figure 4: (a) Vertex-defining umbrella of $X$ with $\mathop{\mathrm{vf}}\nolimits(X)=(3,1)$, (b) corresponding arc-coloured subgraph in $\mathop{\mathrm{\mathcal{F}}}\nolimits(X)$
  • Figure 5: (a) A mono-coloured vertex-defining umbrella of a simplicial surface $X$ with $\mathop{\mathrm{vf}}\nolimits(X)=(2,2)$, (b) the corresponding arc-coloured subgraph in the face graph $\mathop{\mathrm{\mathcal{F}}}\nolimits(X)$
  • ...and 12 more figures

Theorems & Definitions (70)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7: grunbaum1969conjecture
  • Definition 2.8
  • ...and 60 more