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5 - How symmetric can maps on surfaces be?

Published online by Cambridge University Press:  05 July 2013

By
Jozef Širáň
Edited by
Simon R. Blackburn ,
Stefanie Gerke  and
Mark Wildon
Simon R. Blackburn
Affiliation:
Royal Holloway, University of London
Stefanie Gerke
Affiliation:
Royal Holloway, University of London
Mark Wildon
Affiliation:
Royal Holloway, University of London
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Summary

Abstract

A map, that is, a cellular embedding of a graph on a surface, may admit symmetries such as rotations and reflections. Prominent examples of maps with a ‘high level of symmetry’ come from Platonic and Archimedean solids. The theory of maps and their symmetries is surprisingly rich and interacts with other disciplines in mathematics such as algebraic topology, group theory, hyperbolic geometry, the theory of Riemann surfaces and Galois theory.

In the first half of the paper we outline the fundamentals of the algebraic theory of regular and orientably regular maps. The second half of the article is a survey of the state-of-the-art with respect to the classification of such maps by their automorphism groups, underlying graphs, and supporting surfaces. We conclude by introducing the notion of ‘external symmetries’ of regular maps, going well beyond automorphisms, and discuss the corresponding ‘super-symmetric’ maps.

Introduction

Groups are often studied in terms of their action on the elements of a set or on particular objects within a structure. Examples of such situations are abundant and we mention here just a few. Since Cayley's time we know that every group can be viewed as a group of permutations on a set. The study of group actions on vector spaces gave rise to the vast area of representation theory. Investigation of automorphism groups of field extensions generated challenges such as the Inverse Galois Problem. In low-dimensional topology, group actions on trees and on graphs in general led to important findings regarding growth of groups.

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Surveys in Combinatorics 2013 , pp. 161 - 238
Publisher: Cambridge University Press
Print publication year: 2013

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