Random Cayley maps for groups generated by involutions (Q1306313)

Given a generating set \(\Delta\) for a finite group \(\Gamma\), an orientable 2-cell imbedding of the Cayley graph \(G_\Delta(\Gamma)\) is a Cayley map if the rotation of arcs leaving each vertex is given by a fixed cyclic permutation of the generators and their inverses. (Thus a Cayley map covers an index-one voltage graph imbedding.) In this paper the uniform distribution is imposed on the space of all Cayley maps for \(\Gamma\) and \(\Delta\) fixed, where \(\Delta\) consists entirely of involutions. The minimum, maximum, and expected value of the genus random variable on this sample space are determined for \(\Gamma\) abelian, symmetric, or dihedral (with certain specified generating sets). Dihedral groups are used to show that the difference can be arbitrarily large, between two successive values in the genus distribution for Cayley maps (for fixed \(\Gamma\) and \(\Delta\)); thus the analog of Duke's theorem fails badly for Cayley maps. Moreover, for fixed \(\Gamma\) and \(\Delta\), the probability that a given Cayley map is symmetrical is determined.