Regular combinatorial maps

The classical approach to maps is by cell decomposition of a surface. A combinatorial map is a graph-theoretic generalization of a map on a surface. Besides maps on orientable and non-orientable surfaces, combinatorial maps include tessellations, hypermaps, higher dimensional analogues of maps, and certain toroidal complexes of Coxeter, Shephard, and Grünbaum. In a previous paper the incidence structure, diagram, and underlying topological space of a combinatorial map were investigated. This paper treats highly symmetric combinatorial maps. With regularity defined in terms of the automorphism group, necessary and sufficient conditions for a combinatorial map to be regular are given both graph- and group-theoretically. A classification of regular combinatorial maps on closed simply connected manifolds generalizes the well-known classification of metrically regular polytopes. On any closed manifold with nonzero Euler characteristic there are at most finitely many regular combinatorial maps. However, it is shown that, for nearly any diagram D, there are infinitely many regular combinatorial maps with diagram D. A necessary and sufficient condition for the regularity of rank 3 combinatorial maps is given in terms of Coxeter groups. This condition reveals the difficulty in classifying the regular maps on surfaces. In light of this difficulty an algorithm for generating a large class of regular combinatorial maps that are obtained as cyclic coverings of a given regular combinatorial map is given.