Regular pinched maps

This paper concerns pinched surfaces, also known as pseudosurfaces. A map is a graph G embedded on an oriented pinched surface. An arc of a map is an edge of G with a fixed direction. A regular map is one with a group of orientation-preserving automorphisms that acts regularly on the arcs of a map, i.e., that acts both freely and transitively. We study regular maps on pinched surfaces. We give a relation between a regular map on a pinched surface and a natural corresponding regular map on a surface with the pinch points pulled apart. We give several constructions for regular pinched maps and present a plethora of examples. These include strongly connected maps on pinched surfaces (those that do not have a finite set of disconnecting points), as well as examples formed by gluing other regular maps along a finite set of points. D. ARCHDEACON ET AL. / AUSTRALAS. J. COMBIN. 58 (1) (2014), 16–26 17