Characterisation of geodesic self-dual regular surface triangulations

TL;DR

This paper characterizes all subgroups of a specific triangle group that correspond to geodesic self-dual regular surface triangulations, providing a complete enumeration for degrees less than 10, advancing understanding of symmetrical surface triangulations.

Contribution

It offers a complete characterization and enumeration of subgroups related to geodesic self-dual regular triangulations for degrees under 10, a novel classification in this area.

Findings

Complete subgroup characterization for geodesic self-dual triangulations.

Enumeration results for degrees less than 10.

Insight into the symmetry properties of these triangulations.

Abstract

We consider triangulations of closed surfaces in which every vertex is incident to exactly $d$ edges. These triangulations can be identified with subgroups of the triangle group $\langle a,b,c\mid a^2,b^2,c^2,(ab)^3,(ac)^2,(bc)^d\rangle$ that intersect $\langle a,b\rangle$, $\langle a,c\rangle$, and $\langle b,c\rangle$ trivially. The term geodesic duality refers to an external symmetry introduced by Wilson in 1979. Our main result is the characterisation of all subgroups corresponding to geodesic self--dual regular triangulations, together with a complete enumeration for $d < 10$.