Slovak Academy of Sciences

Preface Regular maps and hypermaps are cellular decompositions of closed sur-faces exhibiting the highest possible number of symmetries. The five Platonic solids present the most familiar examples of regular maps. The great dodecahedron, a 5-valent pentagonal regular map on the surface of genus 5 discovered by Kepler, is probably the first known non-spherical regular map. Modern history of regular maps goes back at least to Klein (1878) who described in [109] a regular map of type (3, 7) on the orientable surface of genus 3. In its early times, the study of regular maps was closely connected with group theory as one can see in Burnside’s famous monograph [37], and more recently in Coxeter’s and Moser’s book [53] (Chapter 8). The present-time interest in regular maps extends to their connection to Dyck’s triangle groups, Riemann surfaces, algebraic curves, Galois groups and other areas. Many of these links are nicely surveyed in the papers of Jones [99] and Jones and Singerman [98]. In the present booklet we develop a theory of maps and hypermaps. Symmetries of maps from several points of view are investigated. This booklet can serve as an introductory course to the combinatorial the-ory of maps and hypermaps. It is equipt with exhaustive, but still incomplete list of literature for further reading on the subject. I am grateful to the Combinatorial and Computational Mathematics Center at Pohang University of Science and Technology, Korea, for creating excellent conditions for the work on the book and for accepting it as one of the Lecture Note Series