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Mathematics of Computation

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Archimedean maps of higher genera

Authors: Ján Karabáš and Roman Nedela
Journal: Math. Comp. 81 (2012), 569-583
MSC (2010): Primary 05C30; Secondary 05C10, 05C25
Published electronically: May 13, 2011
MathSciNet review: 2833509
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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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Additional Information

Ján Karabáš
Affiliation: Science and Research Institute, Matej Bel University, Ďumbierska 1, 974 11 Banská Bystrica, Slovakia

Roman Nedela
Affiliation: Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia; Mathematical Institute, Slovak Academy of Sciences, Ďumbierska 1, 974 11 Banská Bystrica, Slovakia
MR Author ID: 262779

Keywords: Polyhedron, Archimedean solid, map, surface, group, graph embedding
Received by editor(s): September 14, 2007
Received by editor(s) in revised form: November 4, 2010
Published electronically: May 13, 2011
Additional Notes: Both authors were partially supported by the grants APVV-51-009605 and VEGA 1/0722/08, grants of Slovak Ministry of Education.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.