The genera, reflexibility and simplicity of regular maps

Abstract

This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to $g-1$, where $g$ is the genus, all orientably-regular maps of genus $p+1$ for $p$ prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda–Nedela–Širáň classification of non-orientable regular maps of Euler characteristic $-p$ where $p$ is prime.