Cayley Maps and Group Extensions Seminar

A Cayley map is a highly symmetrical embedding of a Cayley graph in an orientable surface. A regular Cayley map has additional map automorphisms that make the orientation preserving automorphism group of the map transitive on its set of oriented edges. The regularity of a Cayley map has been shown to be equivalent the existence of a skew-morphism of the underlying group. A skew-morphism is a permutation of the set of elements of a group that generalizes the well known concept of a group automorphism. The analogy goes even further and skew-morphisms can be shown to play a similar role in cyclic extensions of groups as do group automorphisms in the semi direct product extensions. The talk is aimed at general mathematical audience. We begin by reviewing some basic results concerning regular Cayley maps (several of which are based on results due to mathematicians from the University of Southampton). The second half of the talk will deal with cyclic group extensions via skew-morphisms.