Coxeter-Petrie Complexes of Regular Maps

Coxeter-Petrie complexes naturally arise as thin diagram geometries whose rankthree residues contain all of the dual forms of a regular algebraic mapM. Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. G. A. Jones and J. S. Thornton have shown that these involutory duality operations generate the symmetric group S3, giving in all six dual forms, and whose source is the outer automorphism group of the infinite triangle group generated by involutions s1, s2, s3, subject to the additional relation s1s3 = s3s1. In fact, this outer automorphism group is parametrized by the permutations of the three commuting involutions s1, s3, s1s3. These involutions together with the involution s2 can be taken to define the nodes of a Coxeter diagram of shape D4 (with the involution s2 at the central node), and when the original map M is regular, there is a natural extension from M to a thin Coxeter complex of rank 4 all of whose rank-3 residues are isomorphic to the various dual forms of M. These are fully explicated in case the original algebraic map is a Platonic map.