Vertex-Transitive Polyhedra of Higher Genus, I

TL;DR

This paper advances the classification of vertex-transitive polyhedra of higher genus in Euclidean 3-space, establishing new symmetry constraints and fully resolving the case of rotational tetrahedral symmetry.

Contribution

It proves that the symmetry group acts simply transitively on vertices and completes the classification for rotational tetrahedral symmetry.

Findings

Genus g>=2 polyhedra are finite in number with rotational Platonic symmetry.

Symmetry group acts simply transitively on vertices.

The unique rotational tetrahedral example is identified.

Abstract

Since Gruenbaum and Shephard's investigation of self-intersection-free polyhedra with positive genus and vertex-transitive symmetry in 1984 the question of complete classification of such objects in Euclidean 3-space has been open. Due to a recent article by Gevay, Schulte, and Wills, we now know that the genus range g>=2 can only admit a finite number of vertex-transitive polyhedra, all with rotational Platonic symmetry. In this article, we show that the symmetry group must also act simply transitively on the vertices. Furthermore, the case of rotational tetrahedral symmetry is settled completely - the unique example is already known.