Characterisation of geodesic self-dual regular surface 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 $⟨a,b,c\mid a^2,b^2,c^2,(ab{)}^3,(ac{)}^2,(bc{)}^d⟩$ that intersect $⟨a,b⟩$, $⟨a,c⟩$, and $⟨b,c⟩$ 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 $d<10$.