Characterisation of geodesic self-dual regular surface triangulations

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 $d<10$.