Archimedean maps of higher genera

Abstract

The paper focuses on the classification of vertex-transitive polyhe-dral maps of genus from 2 to 4. These maps naturally generalise the spherical maps associated with the classical Archimedean solids. Our analysis is based on the fact that each Archimedean map on an orientable surface projects onto a one-or a two-vertex quotient map. For a given genus g ≥ 2 the number of quotients to consider is bounded by a function of g. All Archimedean maps of genus g can be reconstructed from these quotients as regular covers with covering transformation group isomorphic to a group G from a set of g-admissible groups. Since the lists of groups acting on surfaces of genus 2, 3 and 4 are known, the problem can be solved by a computer-aided case-to-case analysis.