11 - The genus of a group

Summary

This chapter surveys the genus of a finite group. Various symmetric embeddings of Cayley graphs are discussed, together with their associated genus parameters and their relationship to group actions on surfaces. Computations for low genus and certain families of groups are given. Particular attention is paid to general results relating the various genus parameters to each other.

Introduction

The (orientable) genus γ (A) of a finite group A is the smallest integer h such that some Cayley graph for A can be embedded in the orientable surface S_h. (Recall that the Cayley graph C(A, X) for a group A with generating set X has vertex-set A and edges between a and ax, for all a ∈ A and x ∈ X.) The term was first introduced by White [50], but similar ideas appear as far back as the late 19th century. Burnside [6] has two chapters on the ‘graphical representation of a group’ that include the determination of all groups of ‘genus’ 0 and 1 (really the strong symmetric genus, in the language of the next section). The early history is mostly in the context of finite groups of conformal automorphisms of Riemann surfaces, and this context continues to play an important role. On the other hand, Burnside also viewed an embedding of a Cayley graph, or more explicitly the faces of such an embedding, as a way of understanding the relations in a group presentation, in the spirit of Dehn [14] a few years later.