Self‐dual and self‐petrie‐dual regular maps

Regular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈 a, b, c | a 2 = b 2 = c 2 = ( ab ) k = ( bc ) m = ( ca ) 2 = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map ( G ; a, b, c ) is self‐dual if the assignment b ↦ b, c ↦ a and a ↦ c extends to an automorphism of G , and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca . In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k . We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012