Conder, Marston D.; Kwon, Young Soo and Širáň, Jozef
(2014).
*Rigidity and Symmetry*, Springer, pp. 87–96.

DOI (Digital Object Identifier) Link: | https://doi.org/10.1007/978-1-4939-0781-6_5 |
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Google Scholar: | Look up in Google Scholar |

## Abstract

Regular maps are embeddings of graphs or multigraphs on closed surfaces (which may be orientable or non-orientable), in which the automorphism group of the embedding acts regularly on flags. Such maps may admit *external symmetries* that are not automorphisms of the embedding, but correspond to combinations of well known operators that may transform the map into an isomorphic copy: duality, Petrie duality, and the ‘hole operators’, also known as ‘taking exponents’. The group generated by the external symmetries admitted by a regular map is the *external symmetry group* of the map. We will be interested in external symmetry groups of regular maps in the case when the map admits both the above dualities (that is, if it has *trinity symmetry*) and all feasible hole operators (that is, if it is *kaleidoscopic*). Existence of finite kaleidoscopic regular maps was conjectured for every even valency by Wilson, and proved by Archdeacon, Conder and Širáň (2010).

It is well known that regular maps may be identified with quotients of extended triangle groups. In other words, these groups may be regarded as ‘universal’ for constructions of regular maps. It is therefore interesting to ask if similar ‘universal’ groups exist for kaleidoscopic regular maps with trinity symmetry. A satisfactory answer, however, is likely to be very complex, if indeed feasible at all. We demonstrate this (and other things) by a construction of an infinite family of finite kaleidoscopic regular maps with trinity symmetry, all of valency 8, such that the orders of their external symmetry groups are unbounded. Also we explicitly determine the external symmetry groups for the family of kaleidoscopic regular maps of even valency mentioned above.

Item Type: | Conference or Workshop Item |
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Copyright Holders: | 2014 Springer Science+Business Media New York |

ISBN: | 1-4939-0780-8, 978-1-4939-0780-9 |

ISSN: | 1069-5265 |

Keywords: | regular map; group of external symmetries |

Academic Unit/School: | Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics Faculty of Science, Technology, Engineering and Mathematics (STEM) |

Item ID: | 41759 |

Depositing User: | Jozef Širáň |

Date Deposited: | 07 Jan 2015 14:07 |

Last Modified: | 07 Dec 2018 10:28 |

URI: | http://oro.open.ac.uk/id/eprint/41759 |

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