The genera, reflexibility and simplicity of regular maps.

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 (ir-reflexible), 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. [ABSTRACT FROM AUTHOR]