# Regular abelian coverings of cyclic hypermaps

Maureemootoo, Dawn Irene
(2000)
Regular abelian coverings of cyclic hypermaps.
*
University of Southampton, Department of Mathematics,
Doctoral Thesis
*, 165pp.

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## Description/Abstract

The foundations of this work lie in the use of representation theory

to find lattices of regular hypermaps H arising as elementary abelian

coverings of some regular hypermap Ji such that Aut(Tt) = CR. It is

assumed that Aut(7f) = C™ X CR a spUt extension of an elementary

abelian group with cyclic complement. Examples of this situation are

found in [2] where regular orientable imbeddings of the complete graph

Kq, q = pe with p a prime integer, were classified and enumerated. Initial

investigations are of maps obtained from regular orientable imbeddings

of certain highly symmetrical subgraphs Kq of Kq. Results pertaining

to the number and valency of vertices, edges, faces, Petrie polygons, the

genus, the automorphism group and reflexibility are found to be similar

to those obtained in [2]. Subsequently, a description is given of the

general construction. Determination of the lattice of coverings H of H

is found to be dependent upon the signature of the map-subgroup T of

H. Results are obtained for the cases where Tab is a torsion group and

where F"6 is a free abelian group. Hypermap operations induced by outer

automorphisms of the triangle group A(R, R, oo) are also considered

here. The hypermaps are regular of type (R, R,p) and, once again, have

automorphism group C™ XI CR. It was found that when n = 1 these

hypermaps lie in a single orbit under the group of hypermap operations,

whilst when n = 2 they have orbit length <p{R)2 in case R\p - 1 and

length (p(R)2/2 otherwise.

Item Type: | Thesis (Doctoral) |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | University Structure - Pre August 2011 > School of Mathematics |

ePrint ID: | 50625 |

Date Deposited: | 19 Mar 2008 |

Last Modified: | 27 Mar 2014 18:33 |

URI: | http://eprints.soton.ac.uk/id/eprint/50625 |

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