Map operations and \(k\)-orbit maps (Q965221)

A \(k\)-orbit map is a map on a closed surface whose automorphism group has exactly \(k\) orbits on the flags. The paper describes the basic theory of \(k\)-orbit maps and determines all classes of such maps with \(k\leq 4\). Then, exploiting geometric and algebraic operations on maps (such as taking the truncation or medial), the authors enumerate the \(2\)- and \(3\)-orbit maps on the \(2\)-sphere, projective plane, torus, and Klein bottle. Moreover, Hurwitz-like bounds for the order of the automorphism group of \(2\)- and \(3\)-orbit maps are obtained.