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|a2 = b2 = c2 = (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
regular map triangle group self-dual map self-petrie-dual map