Map operations and k-orbit maps

A k-orbit map is a map with k flag-orbits under the action of its automorphism group. We give a basic theory of k-orbit maps and classify them up to k⩽4. “Hurwitz-like” upper bounds for the cardinality of the automorphism groups of 2-orbit and 3-orbit maps on surfaces are given. Furthermore, we consider effects of operations like medial and truncation on k-orbit maps and use them in classifying 2-orbit and 3-orbit maps on surfaces of small genus.