Highly symmetric maps on surfaces with boundary (Q1687947)

A map on a surface is called \textit{regular} if its automorphism group has a single orbit on flags (incident vertex-edge-face triples), or sometimes if the surface is orientable and the group of orientation-preserving automorphisms has a single orbit on arcs (incident vertex-edge pairs). In recent years such maps have attracted a lot of attention, but almost entirely on surfaces without boundary. In this paper the author considers fully regular maps on surfaces with non-empty boundary. He shows that every flag of such a map must meet the boundary, and hence that the automorphism group is always cyclic or dihedral, and then uses these facts to give a complete classification. In fact it turns out that all such maps lie on the closed disk, and so are quotients of regular maps on the sphere. In particular, there are two mutually dual infinite families, and six sporadic examples. Also he extends this classification to regular hypermaps (which are relevant to the study of algebraic curves over real algebraic number fields), and to arc-transitive maps, and observes that a similar classification of edge-transitive maps on surfaces with non-empty boundary would be unrealistic because of the wide variety of examples that occur in that case.