Regular abelian coverings of cyclic hypermaps

Regular abelian coverings of cyclic hypermaps

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.

Maureemootoo, Dawn Irene

abc7e01e-9525-4907-8f45-7052956a7f4f

25 June 2000

Maureemootoo, Dawn Irene

abc7e01e-9525-4907-8f45-7052956a7f4f

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

Record type:
Thesis
(Doctoral)

## 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.

Text

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## More information

Published date: 25 June 2000

Organisations:
University of Southampton

## Identifiers

Local EPrints ID: 50625

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

PURE UUID: 6949e8cd-fa49-41cf-84f7-847225ba9d42

## Catalogue record

Date deposited: 19 Mar 2008

Last modified: 13 Mar 2019 20:50

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## Contributors

Author:
Dawn Irene Maureemootoo

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