Group actions on graphs, maps and surfaces with maximum symmetry

Abstract

Abstract This is a summary of a short course of lectures given at the Groups St Andrews conference in Oxford, August 2001, on the significant role of combinatorial group theory in the study of objects possessing a high degree of symmetry. Topics include group actions on closed surfaces, regular maps, and finite s -arc-transitive graphs for large values of s . A brief description of the use of Schreier coset graphs and computational methods for handling finitely-presented groups and their images is also given. Introduction Historically there has been a great deal of fascination with symmetry — in art, science and culture. One of the key strengths of group theory comes from the use of groups to measure and analyse the symmetries of objects, whether these be physical objects (in 2 or 3 dimensions), or more purely mathematical objects such as roots of polynomials or vectors or indeed other groups. This is now bearing unexpected fruit in areas such as structural chemistry (with the study of fullerenes for example), and interconnection networks (where Cayley graphs and other graphs constructed from groups often have ideal properties for communication systems). The aim of this paper (and the associated short course of lectures given at the Groups St Andrews 2001 conference in Oxford) is to describe a number of instances of symmetry groups of mathematical objects where the order of the group is as large as possible with respect to the genus, size or type of the object.