The University of Southampton
University of Southampton Institutional Repository

Hypermaps: constructions and operations

Hypermaps: constructions and operations
Hypermaps: constructions and operations
It is conjectured that given positive integers l, m, n with l-1 + m-1 + n-1 < 1
and an integer g ≥ 0, the triangle group Δ = Δ (l, m, n) = ⟨X,Y,Z|X l = Y m =
Z n = X Y Z = 1⟩ contains infinitely many subgroups of finite index and of genus
g. This conjecture can be rewritten in another form: given positive integers l,
m, n with l¡1 +m¡1 +n¡1 < 1 and an integer g ≥ 0, there are infinitely many
nonisomorphic compact orientable hypermaps of type (l, m, n) and genus g.
We prove that the conjecture is true, when two of the parameters l, m, n are
equal, by showing how to construct those hypermaps, and we extend the result
to nonorientable hypermaps.

A classification of all operations of finite order in oriented hypermaps is
given, and a detailed study of one of these operations (the duality operation)
is developed. Adapting the notion of chirality group, the duality group of
H can be defined as the minimal subgroup D(H) ≤¦ M on (H) such that
H = D (H) is a self-dual hypermap. We prove that for any positive integer d,
we can find a hypermap of that duality index (the order of D (H) ), even when
some restrictions apply, and also that, for any positive integer k, we can find a
non self-dual hypermap such that |Mon (H) | = d = k. We call this k the duality
coindex of the hypermap. Links between duality index, type and genus of a
orientably regular hypermap are explored.

Finally, we generalize the duality operation for nonorientable regular hypermaps
and we verify if the results about duality index, obtained for orientably
regular hypermaps, are still valid.
Pinto, Daniel Alexandre Peralta Marques
5381c56e-5840-4dea-bb69-f218e9ff1c9a
Pinto, Daniel Alexandre Peralta Marques
5381c56e-5840-4dea-bb69-f218e9ff1c9a
Jones, Gareth
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5

Pinto, Daniel Alexandre Peralta Marques (2009) Hypermaps: constructions and operations. University of Southampton, School of Mathematics, Doctoral Thesis, 164pp.

Record type: Thesis (Doctoral)

Abstract

It is conjectured that given positive integers l, m, n with l-1 + m-1 + n-1 < 1
and an integer g ≥ 0, the triangle group Δ = Δ (l, m, n) = ⟨X,Y,Z|X l = Y m =
Z n = X Y Z = 1⟩ contains infinitely many subgroups of finite index and of genus
g. This conjecture can be rewritten in another form: given positive integers l,
m, n with l¡1 +m¡1 +n¡1 < 1 and an integer g ≥ 0, there are infinitely many
nonisomorphic compact orientable hypermaps of type (l, m, n) and genus g.
We prove that the conjecture is true, when two of the parameters l, m, n are
equal, by showing how to construct those hypermaps, and we extend the result
to nonorientable hypermaps.

A classification of all operations of finite order in oriented hypermaps is
given, and a detailed study of one of these operations (the duality operation)
is developed. Adapting the notion of chirality group, the duality group of
H can be defined as the minimal subgroup D(H) ≤¦ M on (H) such that
H = D (H) is a self-dual hypermap. We prove that for any positive integer d,
we can find a hypermap of that duality index (the order of D (H) ), even when
some restrictions apply, and also that, for any positive integer k, we can find a
non self-dual hypermap such that |Mon (H) | = d = k. We call this k the duality
coindex of the hypermap. Links between duality index, type and genus of a
orientably regular hypermap are explored.

Finally, we generalize the duality operation for nonorientable regular hypermaps
and we verify if the results about duality index, obtained for orientably
regular hypermaps, are still valid.

Text
Daniel_Pinto_PhD.pdf - Other
Download (1MB)

More information

Published date: 2009
Organisations: University of Southampton

Identifiers

Local EPrints ID: 167633
URI: http://eprints.soton.ac.uk/id/eprint/167633
PURE UUID: dfac582a-4d42-448b-ac3a-c85c9d80ac86

Catalogue record

Date deposited: 03 Dec 2010 12:33
Last modified: 14 Mar 2024 02:16

Export record

Contributors

Author: Daniel Alexandre Peralta Marques Pinto
Thesis advisor: Gareth Jones

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×