Regular maps with simple underlying graphs

A regular map is a symmetric embedding of a graph (or multigraph) on some closed surface. In this paper we consider the genus spectrum for such maps on orientable surfaces, with simple underlying graph. It is known that for some positive integers g, there is no orientably-regular map of genus g for which both the map and its dual have simple underlying graph, and also that for some g, there is no such map (with simple underlying graph) that is reflexible. We show that for over 83% of all positive integers g, there exists at least one orientably-regular map of genus g with simple underlying graph, and conjecture that there exists at least one for every positive integer g.