Journal of the European Mathematical Society
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The genera, reflexibility and simplicity of regular mapsMarston D. E. Conder, Jozef Širáň and Thomas W. Tucker (1) Department of Mathematics, University of Auckland, P.O. BOX 92019, 1142, AUCKLAND, NEW ZEALAND
(2) Mathematics Department, Colgate University, NY 13346, HAMILTON, UNITED STATES
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.
Keywords: Regular map, symmetric graph, embedding, genus, chiral, reflexible
Conder Marston, Širáň Jozef, Tucker Thomas: The genera, reflexibility and simplicity of regular maps. J. Eur. Math. Soc. 12 (2010), 343-364. doi: 10.4171/JEMS/200