Archimedean maps of higher genera

Karabáš, Ján; Nedela, Roman

doi:10.1090/S0025-5718-2011-02502-0

Archimedean maps of higher genera
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by Ján Karabáš and Roman Nedela PDF

Math. Comp. 81 (2012), 569-583 Request permission

Abstract:

The paper focuses on the classification of vertex-transitive polyhedral 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 $g\geq 2$ 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 $\mathrm {G}$ 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.

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