Introduction

Symmetric maps – graphs embedded on surfaces with a sufficient ‘level of symmetry’ – have been extensively studied over the last 100 years. Their roots, however, go much deeper, to the Platonic solids of the ancient Greeks and (much later) to Kepler's stellated polyhedra.

The theme of this chapter is how maps can be treated as purely algebraic structures. This is not just an exercise in abstract nonsense: such a viewpoint has historical origins, not only in Coxeter and Moser [11] but also in Edmonds' original rotation schemes [13] and Tutte's flags [42], both of which view maps as permutation groups. Moreover, this algebraicization has a number of important consequences. It allows the simple description and construction of complicated maps – a group presentation is a remarkably efficient data structure. This in turn makes it possible to list all symmetric maps of moderate size, using computer software for group-theoretic calculations. For example, Conder [5] has determined all regular maps of genus at most 100, and Orbanič [34] has found all edge-transitive maps with at most 1000 edges; such lists have been extraordinarily helpful for detecting general phenomena.