Hipermapas 2-restritamente-regulares de baixo género

Universidade de Aveiro

2007

Departamento de Matemática

Rui Filipe

Alves Silva Duarte

Hipermapas 2-restritamente-regulares de baixo

género

2-restrictedly-regular hypermaps of small genus

DOCUMENTO

PROVISÓRIO

Universidade de Aveiro

2007

Departamento de Matemática

Rui Filipe

Alves Silva Duarte

Hipermapas 2-restritamente-regulares de baixo

género

2-restrictedly-regular hypermaps of small genus

Tese apresentada à Universidade de Aveiro para cumprimento dos requisitos

necessários à obtenção do grau de Doutor em Matemática, realizada sob a

orientação científica do Dr. António João de Castilho Breda d'Azevedo,

Professor Associado do Departamento de Matemática da Universidade de

Aveiro

o júri

presidente

Doutor Carlos Alberto Diogo Soares Borrego

Professor Catedrático da Universidade de Aveiro

vogais

Doutor António Carlos Henriques Guedes de Oliveira

Professor Catedrático da Faculdade de Ciências da Universidade do Porto

Doutor Domingos Moreira Cardoso

Professor Catedrático da Universidade de Aveiro

Doutor António João de Castilho Breda d’Azevedo

Professor Associado da Universidade de Aveiro

Doutor Gareth Jones

agradecimentos

Estou muito grato:

Ao Professor João Breda, pela orientação científica.

À Universidade de Aveiro, em particular ao Departamento de Matemática,

pelas condições que me proporcionou e por me ter concedido dispensa de

serviço docente para a realização deste trabalho.

À Unidade de Investigação Matemática e Aplicações (UIMA) o apoio financeiro

concedido.

Aos meus amigos.

À minha família por todo o apoio que me deu, e pelo investimento na minha

formação.

palavras-chave

Hipermapa, restritamente-regular, 2-restritamente-regular,

orientavelmente-regular, pseudo-orientavelmente-orientavelmente-regular, bipartido-orientavelmente-regular, grupo de

quiralidade, índice de quiralidade.

resumo

Nesta tese consideramos hipermapas com grande número de automorfismos

em superfícies de baixo género, nomeadamente a esfera, o plano projectivo, o

toro e o duplo toro.

É conhecido o facto de que o número de automorfismos ou simetrias de um

hipermapa H é limitado pelo seu número de flags, que, genericamente falando,

são triplos vértice-aresta-face mutualmente incidentes. De facto, o número de

automorfismos de H divide o número de flags de H. Hipermapas para os quais

este limite é atingido são chamados regulares e estão classificados nas

superfícies orientáveis até género 101 e em superfícies não-orientáveis até

genero 202, usando computadores.

Neste trabalho classificamos os hipermapas 2-restritamente-regulares na

esfera, no plano projectivo, no toro e no duplo toro, isto é, hipermapas cujo

número de automorfismos é igual a metade do número de flags, e calculamos

os seus grupos quiralidade e índices de quiralidade, que podem ser vistos

como medidas algébricas e numéricas de quanto H se distancia de ser regular.

Estes hipermapas são uma generalização dos hipermapas quirais.

Também introduzimos alguns métodos para construir hipermapas bipartidos.

Duas destas construções têm um papel muito importante no nosso trabalho.

keywords

Hypermap, restrictedly-regular, 2-restrictedly-regular, orientably-regular,

pseudo-orientably-regular, bipartite-regular, chirality group, chirality index.

abstract

This thesis deals with hypermaps having large automorphism group on

surfaces of small genus, namely the sphere, the projective plane, the torus and

the double torus.

It is well-known that the number of automorphisms or symmetries of a

hypermap H is bounded by its number of flags, which are, roughly speaking,

incident triples vertex-edge-face. In fact, the number of automorphisms of H

divides the number of flags of H. Hypermaps for which this upper bound is

attained are called regular and have been classified on orientable surfaces up

to genus 101 and on non-orientable surfaces up to genus 202, using

computers.

In this work we classify the 2-restrictedly-regular hypermaps on the sphere, the

projective plane, the torus and the double torus, that is, hypermaps whose

number of automorphism is equal to half the number of flags and compute their

chirality groups and chirality indices, which may be regarded as algebraic and

numerical measures of how far H deviates from being regular. These

hypermaps are a generalization of chiral hypermaps.

We also introduce some methods for constructing bipartite hypermaps. Two of

those constructions will play an important role in our work.

Introdu tion 1

1 Hypermaps 5

1.1 Denitions andnotations . . . 5

1.2 Thetriangle group . . . 7

1.3 Hypermap subgroups . . . 8

1.4 TheEuler formulaand the Hurwitz bound . . . 13

1.5 Duality . . . 15

1.6 Constru tingbipartite hypermaps. . . 17

1.6.1 The

Walsh

onstru tion . . . 19

1.6.2 The

Pin

onstru tion . . . 21

1.7 Theoperator

Orient

. . . 23

1.8 The losure over and the overing ore . . . 25

1.9 Chiralitygroups and hiralityindi es . . . 26

1.10 Bipartite-reg ular hypermaps . . . 29

2 Hypermapson the sphere 31 2.1 Uniformhypermaps onthe sphere. . . 31

2.2 Bipartite-uniformhypermapsonthe sphere . . . 32

2.3 Chirality groups and hirality indi es of the 2-restri tedly-reg ular hypermaps onthe sphere . . . 33

3 Hypermapson the proje tive plane 39 3.1 Uniformhypermaps onthe proje tive plane . . . 39

3.2 Bipartite-uniformhypermapsonthe proje tive plane . . . 41

3.3 Chirality groups and hirality indi es of the 2-restri tedly-reg ular hypermaps onthe proje tive plane . . . 42

4 Hypermapson the torus 47 4.1 Uniformhypermaps onthe torus . . . 47

4.1.1 Uniformmapson the torus oftypes

(4, 2, 4)

and

(6, 2, 3)

. . . 49

4.1.2 Uniformhypermaps onthe torus oftype

(3, 3, 3)

. . . 62

4.2 Bipartite-uniformhypermapsonthe torus . . . 66

4.3 Chirality groups and hirality indi es of the 2-restri tedly-reg ular hypermaps onthe torus . . . 66

4.3.2 Chirality groups and hirality indi es of the pseudo-orientably-regular hypermapson the torus . . . 70 4.3.3 Chiralitygroupsand hiralityindi esofthebipartite-regularhypermaps

on the torus . . . 71 4.4 A noteon restri tedly-regul ar hypermapson the Kleinbottle . . . 79

5 Hypermaps on the double torus 83

5.1 Regular and orientably-regu lar hypermaps onthe double torus . . . 83 5.2 Pseudo-orientably-regular andbipartite-re gular hypermapson the doubletorus 86 5.3 Chirality groups and hirality indi es of the 2-restri tedly-reg ular hypermaps

on the doubletorus . . . 92 5.3.1 Chiralitygroupsand hiralityindi esofthebipartite-regularhypermaps

on the doubletorus obtained bythe

Walsh

or

Pin

onstru tions . . . 92 5.3.2 Chiralitygroupsand hiralityindi esofthe

∆

+0ˆ

0

-regularhypermapson thedoubletoruswhi harenotobtainedbythe

Walsh

or

Pin

onstru tions 94

A Normal losures, ores and homomorphisms 97

Bibliography 99

Thisthesisdealswithhypermapshavinglargeautomorphismgrouponsurfa esofsmallgenus, namelythe sphere, the proje tive plane, the torus andthe double torus.

Topologi al ly, a hypermap

H

is a ellular imbedding of a onne ted hypergraph

G

into a ompa tsurfa e

S

. When

G

isagraph,wesaythat

H

isamap. TheEuler hara teristi and the genus of

H

arethe Euler hara teristi and the genus of

S

. Roughly speaking, the ags of

H

are its in ident triples vertex-edge-fa e, and a symmetryor an automorphism of

H

is a permutation of the set

Ω

H

of ags of

H

preserving in iden e. The set of all automorphisms ofa hypermap

H

forms apermutation group,

Aut(

H)

, a tingon the set ofagsof

H

. It has been shown [24℄ that every nite group is the group of automorphisms of a map (and hen e ofa hypermap). Thenumber ofautomorphisms of ahypermap

H

isbounded bythe number of ags of

H

, sin e every automorphism is uniquely determined by its ee t on a ag. In addition, the number of automorphisms of

H

divides the number of ags of

H

. Hypermaps forwhi hthisupperboundisattained are alled regular. Regularhypermapsmaybethought of as a generalization of the Platoni solids. When

S

is orientable,

H

is said orientable and the number of automorphisms of

H

whi h indu eorientation-pr eservin g automorphisms of

S

is at most half the number of ags of

H

. When the equalityholds, the hypermap

H

is said orientably-regular. If

H

isorientably-regu lar but not regular, then

H

is hiral.

Algebrai ally,ahypermap

H

is ompletelydeterminedbyahypermapsubgroup

H

,whi h is a subgroup of the free produ t

∆ = C

2

∗ C

2

∗ C

2

. The number of ags of

H

is equal to the indexof

H

in

∆

, andits automorphismgroup is isomorphi to

N

∆

(H)/H

, where

N

∆

(H)

denotes the normalizer in

∆

of

H

. The hypermap

H

is regular if

H

is normal in

∆

, and is orientably-regular if

H

is normal in

∆

+

, one of the seven normal subgroups of

∆

ofindex 2. Following [8℄, we saythata hypermap is2-restri tedly-reg ular ifthe normalizer

N

∆

(H)

in

∆

ofahypermap subgroup

H

isoneofthosesevensubgroupsof

∆

. Inotherwords,ahypermap is2-restri tedly-reg ular ifand onlyifitsgroup ofautomorphisms a tson thesetof agswith 2orbits. Thesehypermaps anbeviewasageneralizationof hiralor irreexiblehypermaps. For further reading on maps and hypermaps see [49, 45, 33, 28, 41, 46, 48, 13, 65℄, see also [23,25, 26, 27℄ for the orientable ase,and [16, 39℄ for maps andhypermapswith boundary.

The lassi ation ofallmaps orhypermapswhi hsatisfya ertain onditionisa ommon probleminmapandhypermaptheory. Regular,orientably-regularand hiralmapsand hyper-mapshavebeen lassied a ording to genusor Euler hara teristi [11,12℄,number ofedges orfa es [70, 7, 10, 77, 54, 69, 40℄,or automorphismgroup [14℄. Edge-transiti ve maps,thatis, mapswhose automorphism group a tstransitively onthe set ofedgeshave been lassied on thesphere(GrünbaumandShephard[37℄)andonthetorus(’irá¬,Tu ker andWatkins[66℄). Anotherproblemis thedeterminati on ofall

g

forwhi h thereisa maporhypermapof genus

g

with a ertainproperty[21, 78℄.

Throughoutthelast entury,manyauthors(Brahana[3℄,Threlfall[62℄,Sherk[55℄,Coxeter and Moser [33℄, Garbe [35℄, Bergau and Garbe [2℄) worked on the lassi ation of regular and orientably-regular maps without the help of omputers. They all ontributed to the lassi ation of regular maps on orientable surfa es up to genus 7 and on non-orientab le surfa es up to genus 8. The generalization to hypermaps was done by Corn and Singerman [28℄, Breda and Jones [15℄ and Breda [7℄ on orientable surfa es up to genus 2 and on non-orientable surfa es up to genus 4. It is well-known that the lassi ation of regular maps and hypermaps on a non-orientable surfa e of genus

g

an be derived from the lassi ation of regular maps and hypermaps on the orientable surfa e of genus

g

− 1

. Chiral maps were studied by Sherk[56℄, Garbe [35℄ and Wilson [75℄. Breda and Nedela [11℄ lassied all hiral hypermapson surfa esupto genus 4. Analmost omplete lassi ation ofregular and hiral maps up to 100 edges an be foundin [70, 69℄. In [19℄,Conder and Dob sányi give omplete listsofallregularand hiralmapsonorientablesurfa esofgenus2to15,andallregularmaps on non-orientable surfa es of genus 4 to 30 (that is, all regular and hiral maps on surfa es with Euler hara teristi between -28 and -2). More re ently, Conder [17℄ obtained lists of regular and hiral maps and hypermapson orientable surfa es of genus 2 to 101 and regular maps and hypermaps on non-orientable surfa es of genus 2 to 202, up to isomorphism and duality, with the helpof the newLowIndexNormalSu bgroups routine in MAGMA[1℄.

Inthisthesiswedetermine,uptoduality,all(isomorphism lassesof)2-restri tedly-reg ular hypermapson the sphere,the proje tive plane, the torus and the doubletorus, and ompute their hiralitygroups and hiralityindi es(see [6℄).

In Chapter 1 we introdu e the basi notation used throughout the text. We present methods for onstru tion bipartite maps. Two of these onstru tions,

Walsh

and

Pin

, will play an important role in our thesis. The rst is indu ed by Walsh's orresponden e [67℄ between hypermapsand bipartitemaps onthe same surfa e. Wealso studythe properties of the orientable double over ofa non-orientable hypermap

H

, whi h isthe smallest orientable hypermap overing

H

(see [13℄).

Chapter 2 deals with 2-restri tedly-reg ular hypermaps on the sphere. Using the Euler formula,weseethatthereisaninnitenumberofpossibilitiesforthevalen iesoftheverti es, edges and fa es of a regular or 2-restri tedly-regular hypermap on the sphere. In ea h ase, there isexa tlyone regular or2-restri tedly-reg ular hypermapwith thosevalen ies. Weshow that all2-restri tedly-reg ular hypermapson the sphere areobtained fromregular hypermaps on the sphere using the

Walsh

or

Pin

onstru tions. Most of the ontent of this hapter is published in [9℄.

Chapter3dealswith hypermapsontheproje tiveplane. Wedetermine the 2-restri tedly-regular hypermapsonthe proje tiveplanebyinspe tingtheregularand2-restri tedly-reg ular hypermaps on the sphere. As on the sphere, all 2-restri tedly-reg ular hypermaps on the proje tiveplaneareobtainedfromregular hypermapsontheproje tiveplaneusingthe

Walsh

or

Pin

onstru tions. There is an innite number of possibilities for the valen ies of the verti es, edges and fa es of a regular or 2-restri tedly-regular hypermap on the proje tive plane. Inea h ase,thereisatmostoneregularor2-restri tedly-reg ularhypermapwiththose valen ies.

Hypermaps on the torus are studied in Chapter 4. Our main referen es are the work of Singerman andSyddall[57, 58℄ onuniform maps,andthe workofCoxeterandMoser[33℄ on orientably-regular maps. Onthe torus,the Eulerformulagivesanitenumber ofpossibilities forthevalen iesoftheverti es,edgesandfa esofaregularor2-restri tedly-reg ularhypermap,

andinea h asethereisaninnitenumberofnon-isomorphi regularand2-restri tedly-reg ular hypermapswith thosevalen ies. Itisshownthatthe 2-restri tedly-reg ular hypermapson the torus are either uniform or obtained from regular hypermaps on the torus using the

Walsh

and

Pin

onstru tions. Wealsointrodu e anotation forthe uniformhypermapsonthetorus. Finally,in Chapter5,we lassifyall2-restri tedly-reg ular hypermapsonthe doubletorus. Ourwork in this Chapterwasinuen ed by[15℄.

Hypermaps

Inthis hapterwe introdu ebasi terminology fromthe theoryofhypermapsandat thesame time establishour notation.

1.1 Denitions and notations

A hypermap is a four-tuple

H = (Ω

H

, h

0

, h

1

, h

2

)

where

h

0

, h

1

, h

2

arepermutation s of a non-empty set

Ω

H

su h that

h

0

2

_{= h}

1

2

= h

2

2

= 1

and

hh

0

, h

1

, h

2

i

is transitive on

Ω

H

. The elements of

Ω

H

are alled ags of

H

, the permutation s

h

0

,

h

1

and

h

2

are alled anoni al generators of

H

and the group

Mon(

H) = hh

0

, h

1

, h

2

i

is the monodromy group of

H

. One says that

H

is a map if

(h

0

h

2

)

2

_{= 1}

. A hypermap is said nite if its set of ags is nite. If the permutation s

h

0

,

h

1

and

h

2

are xed-point free, we saythat

H

has no boundary or that

H

is a hypermap without boundary. Hen eforth, all hypermaps areto be nite and without boundary unlessotherwisespe ied.

The hyperverti es or

0

-fa es of

H

orrespond to

hh

1

, h

2

i

-orbits on

Ω

H

. Likewise, the hyperedges or

1

-fa es and hyperfa es or

2

-fa es orrespond to

hh

0

, h

2

i

- and

hh

0

, h

1

i

-orbits on

Ω

H

, respe tively. Ifaag

ω

belongstotheorbitdetermininga

k

-fa e

f

wesaythat

ω

belongs to

f

,orthat

f

ontains

ω

. Weusethetermsverti es,edgesandfa es insteadofhyperverti es, hyperedges and hyperfa es, for short. We denote the numbers of verti es, edgesand fa es of

H

by

V (

H)

,

E(

H)

and

F (

H)

. When justone hypermap, say

H

, isunder dis ussion, we omit the letter

H

fromhypermap-theoreti symbols andwrite, forinstan e

Ω

,

V

,

E

and

F

instead of

Ω

H

,

V (

H)

,

E(

H)

and

F (

H)

.

Let

{i, j, k} = {0, 1, 2}

. Wesaythatthe

k

-fa e

f = ω

hR

i

, R

j

i

andthe

j

-fa e

e = σ

hR

i

, R

k

i

arein ident if

f

∩ e 6= ∅

. Inother words, in iden e isgiven bynon-emptyinterse tion. Two

k

-fa es

f

and

f

′

are adja ent if both are in ident to a

j

-fa e

g

. The valen y of a

k

-fa e

f = w

_hh

i

, h

j

i

(of a nite hypermap without boundary), where

ω

∈ Ω

H

, is the leastpositive integer

n

su h that

(h

i

h

j

)

n

_{∈ Stab(w)}

. Sin e

h

i

2

_{= h}

j

2

= 1

and

h

i

and

h

j

are xed-point free,

f

has

2n

elements, so the valen y of a

k

-fa e is equal to half of its ardinality. If, for ea h hoi e of indi es

i, j

∈ {0, 1, 2}

, all

hh

i

, h

j

i

-orbits on

Ω

H

have the same ardinality, we saythat

H

isuniform. Whenallverti es,edgesandfa esof

H

havevalen ygreater thanone, we anthink ofa agasan in ident vertex-edge-fa e triple

(v, e, f )

. Ahypermap

H

hastype

(l, m, n)

if

l

,

m

and

n

are the least ommon multiples of the valen ies of the verti es, edges and fa es,respe tively. In otherwords, the type of a hypermap

H

is

(l

0

, l

1

, l

2

)

if

l

i

,

l

j

and

l

k

arethe ordersof

h

j

h

k

,

h

k

h

i

and

h

i

h

j

. When

H

isuniform,

H

hastype

(l, m, n)

ifandonly if

l

,

m

and

n

arethe valen ies ofthe verti es,edgesand fa esof

H

, respe tively.

Topologi ally,mapsandhypermaps anberepresentedby ellularimbeddingsof onne ted graphs and hypergraphs into ompa t surfa es. A map

M

an be represented by a ellular imbedding of a onne ted graph

G

into a ompa t surfa e

S

, where the verti es, edges and fa es of the imbedding orrespond to the verti es, edges and fa es of

M

. Using the well-known orresponden eofWalshbetween hypermapsand bipartitemaps des ribed in[67℄,we an representahypermapbya ellularimbeddingofabipartitegraph (thatis, ahypergraph)

G

into a ompa t surfa e

S

, where the verti es of

G

orrespond to the verti es and edges of

H

and two verti es of

G

are onne ted by an edge if and only if they form an in ident pair vertex-edgeof

H

.

Alternatively, a hypermap

H

an be represented by a ellular imbedding of a onne ted trivalent graph

G

into a ompa t surfa e

S

, together with a labelling of the fa eswith labels 0, 1and2 sothatea hedge of

G

isin ident with two fa es arrying dierent labels. Inother words,

H

anberepresented bytheS hreier (right) osetgraph (seeŸ3.7of[33℄,Ÿ7. of[64℄ or Ÿ4-3. of[68℄)forthe stabilizerof aag

ω

∈ Ω

H

in the monodromygroup of

H

,

Mon(

H)

,with respe t to the generators

h

0

,

h

1

and

h

2

, with free edges repla ing loops. The verti es of the graph

G

orrespond to the ags of

H

and the fa es labelled with

k

orrespond to the

k

-fa es of

H

.

When

H

is represented bya ellular imbeddingofa onne ted hypergraph

G

ona surfa e

S

, we saythat

G

is the underlying hypergraph of

H

and that

S

is the underlying surfa e of

H

. A hypermap

H

has no boundary when its underlying surfa e

S

has no boundary. The Euler hara teristi and the genus of a hypermap

H

are the Euler hara teristi and the genus of its underlying surfa e

S

, respe tively. We speakof hara teristi of

H

, meaning the Euler hara teristi of

H

,for short. Hypermapsimbedded onthe sphere are alled spheri al; hypermapsimbedded on the torus are alled toroidal.

A overing fromahypermap

H = (Ω

H

, h

0

, h

1

, h

2

)

toanotherhypermap

G = (Ω

G

, g

0

, g

1

, g

2

)

is afun tion

ψ : Ω

H

→ Ω

G

that ommutes a ording to the following diagram:

Ω

H

h

i

_//

ψ



Ω

H

ψ



Ω

G

g

i

_//

Ω

G

,

that is, su h that

h

i

ψ = ψg

i

for all

i

∈ {0, 1, 2}

. Sin e

Mon(

G)

a ts transitively on

Ω

G

,

ψ

is surje tive. Be ause

Mon(

H)

a tstransitivelyon

Ω

H

,the overing

ψ

is ompletelydetermined by the image of a ag of

H

. By von Dy k's theorem ([42℄, p. 28) the assignment

h

i

7→ g

i

extends to a group epimorphism

Ψ : Mon(

H) → Mon(G)

alled the anoni al epimorphism. The overing

ψ

isanisomorphism ifitisinje tive. Ifthereisa overing

ψ

from

H

to

G

,wesay that

H

overs

G

or that

G

is overed by

H

, andwrite

H → G

; if

ψ

isan isomorphism we say that

H

isisomorphi to

G

, or that

H

and

G

are isomorphi , and write

H ∼

=

G

. When

ψ

is a overingfrom

H

to

G

and

|Ω

H

| = 2|Ω

G

|

wesaythat

ψ

isadouble overing. Anautomorphism or a symmetry of

H

is an isomorphism

ψ : Ω

H

→ Ω

H

from

H

to itself, that is, a fun tion

ψ

that ommutes with the anoni al generators. Naturally, the set of all automorphisms (or symmetries) of

H

forms agroup under omposition, alled theautomorphism group of

H

and denoted by

Aut(

H)

. Sin e for all

ω

∈ Ω

,

(ω

hh

i

, h

j

i)ψ = ωψhg

i

, g

j

i

, a overing

ψ : Ω

H

→ Ω

G

indu es a surje tive mapping between the set of

k

-fa es of

H

and the set of

k

-fa es of

G

; an isomorphism indu esabije tive orresponden ebetween thesetof

k

-fa es of

H

and thesetof

k

-fa esof

G

. Anautomorphism

ψ

is alledaree tion ifthereisaag

ω

∈ Ω

and

k

∈ {0, 1, 2}

su hthat

ωψ = ωr

k

.

Using the Eu lidean Division Algorithm,one an easily showthe following result.

Lemma 1.1.1. Let

ψ : Ω

H

→ Ω

G

be a overing from

H

to

G

and

ω

∈ Ω

H

. Thenthe valen y of the

k

-fa e of

G

ontaining

ωψ

dividesthe valen y of the

k

-fa e of

H

ontaining

ω

.

1.2 The triangle group

Thefreeprodu t

∆ = C

2

∗ C

2

∗ C

2

=

hR

0

, R

1

, R

2

| R

0

2

= R

1

2

= R

2

2

= 1

i

is alled the triangle group. By the torsion theorem for free produ ts (Theorem 1.6 in ŸIV.1 of[51℄), the onjugatesof

R

0

,

R

1

and

R

2

arethe onlynon-identityelementsof niteorder in

∆

. More generally, for ea h triple

(l, m, n)

∈ (N ∪ {∞})

3

, the extended triangle group is the group

∆(l, m, n) =

_hR

0

, R

1

, R

2

| R

0

2

= R

1

2

= R

2

2

= (R

1

R

2

)

l

= (R

2

R

0

)

m

= (R

0

R

1

)

n

= 1

i

where we regardequations of the form

(R

i

R

j

)

∞

_{= 1}

asbeingva uous.

For positive integers

l, m, n

, the extendedtrianglegroup

∆(l, m, n)

isthe group generated byree tionsinthe sidesofatrianglewith angles

π/l

,

π/m

and

π/n

. Thistrianglewilllieon thesphere,theEu lideanplaneorthehyperboli planedependingonwhether

1/l +1/m+1/n

isgreater than, equal to orlessthan 1, respe tively. It iswell-known that:

• ∆(1, m, n) = ∆(1, k, k) ∼

= D

k

, where

k = gcd(m, n)

;

• ∆(2, 2, n) ∼

= D

n

× C

2

;

• ∆(2, 3, 3) ∼

= S

4

;

• ∆(2, 3, 4) ∼

= S

4

× C

2

;

• ∆(2, 3, 5) ∼

= A

5

× C

2

.

If

N

isanormal subgroupof

∆

ofindex2,then

∆/N

,havingorder2,isisomorphi to

C

2

. Consequently,the group

∆

has7subgroups ofindex2 (see[13℄),the kernelsofthe

2

3

_{− 1 = 7}

group epimorphisms

ϕ : ∆

→ C

2

:

∆

+

=

hR

1

R

2

, R

2

R

0

i

∆

=

hR

1

R

2

, R

2

R

0

, R

0

R

1

i,

∆

ˆ

k

=

hR

i

, R

j

i

∆

=

hR

i

, R

j

, R

i

R

k

, R

j

R

k

i,

∆

k

=

hR

k

, R

i

R

j

i

∆

=

hR

k

, R

i

R

j

, R

j

R

k

R

i

i,

where

{i, j, k} = {0, 1, 2}

. The subgroup

∆

+

isoften alled the even subgroup of

∆

.

If

N

is normal subgroup of

∆

of index 4,then

∆/N

, being a group of order 4 generated by ree tions, is

V

4

∼

= C

2

× C

2

. By taking

ϕ : ∆

→ C

2

× C

2

a group epimorphism su hthat

N = ker ϕ

, and

π

1

and

π

2

the proje tions

C

2

× C

2

→ C

2

, one an seethat

N

1

= ker ϕπ

1

and

N

2

= ker ϕπ

2

are normal subgroups of

∆

of index 2 and

N = N

1

∩ N

2

. Consequently, the

normal subgroups of

∆

of index 4 areinterse tions of normal subgroups of

∆

of index 2. By inspe tion we an seethat

∆

has7 normal subgroups of index4 (see [13℄):

∆

012

=

_hR

i

R

j

R

k

i

∆

=

hR

i

R

j

R

k

, R

j

R

k

R

i

, R

k

R

i

R

j

i

= ∆

i

_{∩ ∆}

j

= ∆

0

_{∩ ∆}

1

_{∩ ∆}

2

,

∆

+kˆ

k

=

_hR

i

R

j

, (R

j

R

k

)

2

i

∆

=

hR

i

R

j

, (R

i

R

j

)

R

k

, (R

j

R

k

)

2

, (R

k

R

i

)

2

i

= ∆

+

_{∩ ∆}

k

= ∆

k

_{∩ ∆}

ˆ

k

= ∆

k

ˆ

_{∩ ∆}

+

= ∆

+

_{∩ ∆}

k

_{∩ ∆}

ˆ

k

,

∆

ˆiˆjk

=

hR

k

, (R

i

R

j

)

2

i

∆

=

hR

k

, R

k

R

i

, R

k

R

j

, R

k

R

i

R

j

, (R

i

R

j

)

2

i

= ∆

ˆi

_{∩ ∆}

ˆ

j

= ∆

ˆ

j

_{∩ ∆}

k

= ∆

k

_{∩ ∆}

ˆi

= ∆

ˆi

_{∩ ∆}

ˆ

j

_{∩ ∆}

k

where

{i, j, k} = {0, 1, 2}

. We write

∆

ˆ

01ˆ

2

and

∆

0ˆ

1ˆ

2

insteadof

∆

ˆ

0ˆ

21

and

∆

ˆ

1ˆ

20

, for simpli ity. Let

∆

′

bethederivedgroup(thatis,the ommutatorsubgroup)of

∆

. Forall

i, j

∈ {0, 1, 2}

,

(R

i

R

j

)

2

= [R

i

, R

j

]

∈ ∆

′

, so the rst homology group of

∆

is

∆/∆

′

_∼

_{= C}

2

× C

2

× C

2

and

∆

′

=

_h(R

1

R

2

)

2

, (R

2

R

0

)

2

, (R

0

R

1

)

2

i

∆

= ∆

ˆ

0

∩ ∆

ˆ

1

∩ ∆

2

ˆ

is anormal subgroup of

∆

of index 8.

1.3 Hypermap subgroups

Given a group

G

, we denote by

Z(G)

the enter of

G

. If

H

is a subgroup of

G

, then we denote by

N

G

(H)

,

H

G

and

H

G

, the normalizer, the normal losure and the ore of

H

in

G

, respe tively.

Ea hhypermap

H

givesrisetoatransitivepermutationrepresentation

ρ

H

: ∆

→ Mon(H)

,

R

i

7→ h

i

ofthefreeprodu t

∆ = C

2

∗ C

2

∗ C

2

. Thegroup

∆

a tsnaturally andtransitivelyon

Ω

H

via

ρ

H

. Thestabilizer

H = Stab

∆

(ω)

of aag

ω

∈ Ω

H

under the a tionof

∆

is alledthe hypermap subgroup or fundamental group of

H

. Sin e

∆

a ts transitively on

Ω

H

, hypermap subgroups areuniqueupto onjugation in

∆

. Thevalen yofa

k

-fa e ontaining

ω

istheleast positiveinteger

n

su hthat

(R

i

R

j

)

n

_{∈ Stab}

∆

(ω) = H

;more generally, thevalen yofa

k

-fa e ontaining the ag

σ = ω

· g = ω(g)ρ

H

∈ Ω

H

,where

g

∈ ∆

,istheleastpositiveinteger

n

su h that

(R

i

R

j

)

n

_{∈ Stab}

∆

(σ) = Stab

∆

(ω

· g) = Stab

∆

(ω)

g

= H

g

. We remark that a hypermap of type

(l, m, n)

an be regarded as a transitive permutation representation of the extended triangle group

∆(l, m, n)

(see [13℄).

Lemma 1.3.1. Let

H

and

G

be hypermaps with hypermap subgroups

H

and

G

respe tively. Then

H → G

if andonlyif

H

⊆ G

g

for some

g

∈ ∆

.

Proof. Let

ω

∈ Ω

H

and

σ

∈ Ω

G

su h that

H = Stab

∆

(ω)

and

G = Stab

∆

(σ)

.

(

⇒)

Let

ϕ : Ω

H

→ Ω

G

be a overing and

g

∈ ∆

su h that

ωψ = σg

. Then, for all

h

∈ ∆

,

h

∈ H ⇔ ωh = ω ⇒ σgh = ωψh = ωhψ = ωψ = σg ⇔ h ∈ Stab

∆

(σg) = Stab

∆

(σ)

g

= G

g

,

that is,

H

⊆ G

g

.

(

_⇐)

If

H

⊆ G

g

, then

ϕ : Ω

H

→ Ω

G

,

ωhϕ = σgh

iswelldened and isa overing

H → G

. Corollary 1.3.2. Let

H

and

G

be hypermapswithhypermap subgroups

H

and

G

respe tively. Then

H ∼

=

G

ifand onlyif

H = G

g

for some

g

∈ ∆

. In other words,

H

and

G

are isomorphi if and onlyif there is an innerautomorphism

θ

of

∆

su h that

Hθ = G

.

This last result shows that there is a natural orresponden e between the isomorphism lasses ofhypermapsand the onjugation lassesof subgroups of

∆

.

Let

H

beahypermapsubgroupof

H

. Denoteby

Alg(H) = (∆/

r

H,

·H

∆

R

0

,

·H

∆

R

1

,

·H

∆

R

2

)

where

·H

∆

R

i

: ∆/

r

H

→ ∆/

r

H

,

Hg

7→ HgH

∆

R

i

= HgR

i

. Wesaythat

Alg(H)

is an algebrai presentation of

H

.

Lemma 1.3.3. Let

Alg(H)

be as above. Then

H

is isomorphi to

Alg(H)

. Furthermore, the groups

Mon(

H)

and

∆/H

∆

are isomorphi .

ThisLemmashowsthat,upto isomorphism,everyhypermap

H

is ompletely determined by a hypermap subgroup

H

. For simpli ity, we do not dierentiate

H

from its algebrai presentations, and so we see, for instan e,

Ω

H

as

∆/

r

H

and

Mon(

H)

as

∆/H

∆

, for some hypermap subgroup

H

of

H

.

Lemma1.3.4. Let

H

be ahypermap,

ω

∈ Ω

H

and

H = Stab

∆

(ω)

ahypermapsubgroup of

H

. Then

Aut(

H) ∼

= N

∆

(H)/H

. Moreover,

h

∈ N

∆

(H)

if and only if for every ag

Hg

∈ ∆/

r

H

there is an automorphismof

H

whi h maps

Hg

to

Hhg

.

Notethatanautomorphism

ψ

isa ree tionifand onlyifthereis

g

∈ ∆

and

k

∈ {0, 1, 2}

su hthat

R

k

∈ H

g

.

Ofthetwogroups

Mon(

H)

and

Aut(

H)

,thersta tstransitivelyon

Ω

(bydenition)and the se ond, due to the ommutativity of the automorphisms with the anoni al generators, a tssemi-regularly on

Ω

H

. These twoa tions give rise to the following inequalities:

| Mon(H)| ≥ |Ω

H

| ≥ | Aut(H)|.

(1.1)

Indeed, if

H

is a hypermap subgroup of

H

, then

| Mon(H)| = [∆ : H

∆

]

,

|Ω

H

| = [∆ : H]

and

| Aut(H)| = [N

∆

(H) : H]

.

Lemma 1.3.5. The followingstatements are equivalent:

1.

| Mon(H)| = |Ω

H

|

, that is,

Mon(

H)

a ts regularly on

Ω

H

; 2.

|Ω

H

| = | Aut(H)|

, that is,

Aut(

H)

a tsregularly on

Ω

H

; 3.

H

has a hypermap subgroup whi his normal in

∆

.

If

Mon(

H)

or

Aut(

H)

a tregularlyon

Ω

H

, orequivalently,if

H

hasahypermapsubgroup whi hisnormal in

∆

,then

H

issaidregular. Itiswell-knownthatevery regularhypermapis uniform butthe onverseis nottrue. InChapter4we an nduniform hypermapswhi h are not regular.

Let

H

beahypermapsubgroupofahypermap

H

. Following [8℄,if

H

≤ Θ

forsome

Θ ⊳ ∆

, we saythat

H

is

Θ

- onservative. Wesaythat

H

is

•

orientable if

H

is

∆

+

- onservative,

•

bipartite if

H

is

∆

ˆ

0

- onservative,

•

pseudo-orientabl e if

H

is

∆

0

- onservative 1 . 1

Moreover, given

k

∈ {0, 1, 2}

, we say that

H

is

k

-bipartite if

H

is

∆

ˆ

k

- onservative, and

k

-pseudo-orientabl e if

H

is

∆

k

- onservative. In addition, a

k

-bipartite hypermap is also alled vertex-bipartite if

k = 0

, edge-bipartite if

k = 1

, andfa e-bipartite if

k = 2

.

A hypermap

H

is orientable if and only if its underlying surfa e is orientable. Sin e

∆

+

_{∩ ∆}

ˆi

_{= ∆}

+

_{∩ ∆}

i

_{= ∆}

ˆi

_{∩ ∆}

ˆi

(seeSe tion1.2),anorientablehypermap

H

is

∆

ˆ

k

- onservative ifandonly if

H

is

∆

k

- onservative;a non-orientab lehypermap annotbesimultaneously

∆

ˆ

k

- onservative and

∆

k

- onservative. A hypermap

H

is bipartite if and only if we an divide its set of verti es into two parts sothat onse utive verti es around an edge or a fa e arein alternate parts, that is, if for all

ω

∈ Ω

H

, the verti es ontaining

ω

and

ωh

0

are in dierent parts. A hypermap

H

is pseudo-orientable ifwe an give orientations to the verti es sothat onse utive verti es around an edge or a fa e have dierent orientations, that is, if for all

ω

_{∈ Ω}

H

, the verti es ontaining

ω

and

ωh

0

have dierent orientations.

Lemma 1.3.6. If

H

is bipartite or pseudo-orientab le, then all edges and all fa es have even valen ies. Proof. Let

Θ

be

∆

ˆ

0

or

∆

0

,

H

a

Θ

- onservativehypermap,

ω

∈ Ω

H

, and

H = Stab

∆

(ω)

. If

m

and

n

arethevalen iesoftheedgeandthefa e ontainingtheag

ωg

,then

(R

2

R

0

)

m

_{, (R}

0

R

1

)

n

∈

Stab

∆

(ωg) = H

g

⊆ Θ

g

= Θ

. In both ases

m

and

n

mustbeeven.

Let

Θ

be a normal subgroup of

∆

and

H

a

Θ

- onservativehypermap. An automorphism

ϕ

∈ Aut(H)

is said

Θ

- onservative ifit preserves the

Θ

-orbits on

Ω

H

= ∆/

r

H

, that is, if for all

Hg

∈ ∆/

r

H

,

Hg

and

(Hg)ϕ

arein the same

Θ

-orbit. Sin e

Θ

isa normal subgroupof

∆

ontaining

H

,

Θ

ontains

H

∆

andso

Θ/H

∆

isanormal subgroupof

∆/H

∆

= Mon(

H)

. Sin e every overing is determined bythe image of aag,weget the following result.

Lemma 1.3.7. Let

Θ

be a normal subgroup of

∆

and

H

a

Θ

- onservative hypermap with hypermap subgroup

H

. An automorphism

ϕ

of

H

is

Θ

- onservative if and only if

Hϕ

∈

H

· Θ/H

∆

.

Proof. Only the ne essary ondition needs to be proved. Let

Hϕ = Ht

, with

t

∈ Θ

. Then, for all

g

∈ ∆

,

t

g

_{∈ Θ}

g

_{= Θ}

and

(Hg)ϕ = Hϕg = Htg = Hgt

g

_{∈ Hg · Θ/H}

∆

.

The set of all

Θ

- onservative automorphisms of a

Θ

- onservative hypermap

H

forms a group under omposition denoted by

Aut

Θ

₍

_H)

. The groups of

∆

+

- and

∆

+0ˆ

0

- onservative automorphisms of

H

are alsodenoted by

Aut

+

₍

_H)

and

Aut

+0ˆ

0

₍

_H)

, respe tively.

Now let

Θ

be a normal subgroup of

∆

of index 2. Then every

Θ

- onservative hypermap

H

hasexa tly two

Θ

-orbits. An automorphism

ϕ

of

H

is alled

Θ

-preserving if

ϕ

stabilizes the twoorbits, andis alled

Θ

-reversing if

ϕ

inter hanges thetwoorbits. Wealsosaythatan automorphism

ϕ

ofanorientablehypermapisorientation-preserving if

ϕ

is

∆

+

-reversing,and orientation-reversingif

ϕ

is

∆

+

-reversing. Thegroupoforientation-pr eservin gautomorphisms of an orientable hypermap

H

,

Aut

+

₍

_H)

, is often alled the rotation group of

H

.

When

H ⊳Θ

,

H

is alled

Θ

-regular. If

H

is

Θ

-regularbutnotregular,

H

is alled

Θ

- hiral. We say that

H

is orientably-regular if

H

is

∆

+

-regular, orientably- hiral if

H

is

∆

+

- hiral, bipartite-regu l a r if

H

is

∆

ˆ

0

-regular,bipartite- hiral if

H

is

∆

ˆ

0

- hiral,pseudo-orientably-regul ar if

H

is

∆

0

-regularand pseudo-orientabl y- hiral if

H

is

∆

0

- hiral.

Moregenerally, given

k

∈ {0, 1, 2}

, we saythat

H

is

k

-bipartite-regul a r if

H

is

∆

ˆ

k

-regular,

k

-bipartite- hiral if

H

is

∆

ˆ

k

- hiral,

k

-pseudo-orientably-regul ar if

H

is

∆

k

-regular, and

k

-pseudo-orientabl y- hiral if

H

is

∆

k

hypermap is also alled vertex-bipartite-regular (resp. vertex-bipartite- hiral) if

k = 0

, edge-bipartite-regu lar (resp. edge-bipartite- h iral) if

k = 1

, and fa e-bipartite-regular (resp. fa e-bipartite- hiral) if

k = 2

.

The group of

Θ

- onservative automorphisms of a

Θ

- onservative hypermap

H

,

Aut

Θ

₍

_H)

, isisomorphi to

N

Θ

(H)/H

. When

H

is

Θ

-regular,

N

Θ

(H) = Θ

andso

Aut

Θ

₍

_H)

isisomorphi to

Θ/H

. Thehypermap

H

is

Θ

-regular ifand onlyifits

Θ

- onservativeautomorphism group

Aut

Θ

(

_H)

a ts transitively on ea h

Θ

-orbitin

Ω

H

.

Ahypermap

H

isrotary (see[72℄formaps)ifthereis

ω

∈ Ω

H

and

υ, ϕ

∈ Aut(H)

with the property that

υ

and

ϕ

y li ally permute the onse utive edges in ident to the vertex

v

and the fa e

f

ontaining

ω

, respe tively. In other words, a hypermap isrotary ifthe normalizer in

∆

of a hypermap subgroup ontains

∆

+

. An orientable hypermap

H

isrotary ifand only if

H

is orientably-regular; a non-orientable hypermap

H

is rotary if and only if

H

is regular (see [33, 72℄ for maps). A hypermap

H

is said reexible if its automorphism group has an orientation-reversingautomorphismand hiral orirreexible otherwise([33,49℄). Orientably-regular maps and hypermaps have often been alled regular [3, 33, 28, 25, 26, 27℄, while regular mapsand hypermapshavebeen alled reexible [33℄.

Following[8℄,ahypermap

H

is alledrestri tedly-regular if

H

is

Θ

-regularforsomenormal subgroup

Θ

with niteindex in

∆

. If

H ⊳ Θ

and

Θ ⊳ ∆

, then

H

⊆ Θ ⊆ (N

∆

(H))

∆

⊆ N

∆

(H),

thatis, when

H

isrestri tedly-regul ar , the subgroup

(N

∆

(H))

∆

, alled regularity-subgroup of

H

, isthe largest normal subgroupof

∆

in whi h

H

isnormal.

More generally, we saythat

H

is

k

-restri tedly-regular if

k

is the index of the regularity-subgroupof

H

in

∆

, thatis, if

k = [∆ : (N

∆

(H))

∆

]

. Theindex

k

is alled the restri ted rank of

H

. Sin e

|Ω

H

| = [∆ : H] = [∆ : (N

∆

(H))

∆

]

· [(N

∆

(H))

∆

: H]

= k

· [(N

∆

(H))

∆

: H]

≤ k · [N

∆

(H) : (N

∆

(H))

∆

]

· [(N

∆

(H))

∆

: H]

= k

_{· [N}

∆

(H) : H]

= k

_{· | Aut(H)|,}

when

H

is

k

-restri tedly-regu lar,

|Ω

H

|/| Aut(H)| ≤ k

and

k

| |Ω

H

|

. The restri ted rank of a hypermap

H

an be regarded as a numeri al measure of how far

H

deviates from being regular.

A 1-restri tedly-reg ular hypermap is a regular hypermap; a 2-restri tedly-reg ular hyper-mapisa

Θ

- hiral hypermap, where

Θ

is1 ofthe 7 normal subgroups of

∆

ofindex 2.

Lemma 1.3.8. A hypermap is

2

-restri tedly-regular if and only if the number of automor-phisms of

H

isequal to halfthe number of ags.

In [47℄, Jones alled a map

M

just-edge-tra nsitive if

M

is 4-restri tedly-reg ular and its regularitysubgroup is

∆

ˆ

01ˆ

2

. The lassi ation of

∆

012

-regular hypermaps of small genus, as well astheir hiralitygroups and hiralityindi es an be foundin [5℄.

The typesautomorphismgroups ofedge-transitive maps,whi hin lude all 2-restri tedly-regular maps ex eptthe

∆

ˆ

1

anedge-transitive mapwith anedge-transitive mapwith regularity-automorphismgroup oftype

. . .

automorphismgroup of type

. . .

-subgroups

(Wilson) (Graver &Watkins)

I

1

∆

IIa

2

P

_ex

_∆

+

IIb

2ex

∆

2

2

∗

_ex

_∆

0

II

2

∆

ˆ

0

2

∗

∆

ˆ

2

IId

2

P

_∆

1

IIIa

3

∆

ˆ

01ˆ

2

IIId

5

∆

+0ˆ

0

5

∗

∆

+2ˆ

2

IIIe

5

P

_∆

012

Table 1.1: Corresponden ebetween edge-transitive mapsand restri tedly-regul ar maps.

in [36℄. In Table 1.1 we give the orresponden e between types of edge-transitive maps of Wilson and ofGraver and Watkins,and their regularity-subgroups.

Let

Θ

be a normal subgroup of

∆

. The hypermap with hypermap subgroup

Θ

is alled the trivial

Θ

-hypermap and denoted by

T

Θ

. It is a regular hypermap with

[∆ : Θ]

ags whi h mayhave boundary. InŸ5of [13℄, BredaandJones lassifythe 16trivial

Θ

-hypermaps with abelian automorphism group. Their hypermap subgroups are the 16 normal subgroups of

∆

ontaining

∆

′

(see Se tion 1.2). By Lemma 1.3.1, a hypermap

H

is

Θ

- onservative if and only if

H

overs

T

Θ

. Let

H

be a

Θ

- onservative hypermap,

ϕ

a overing from

H

to

T

Θ

and

{v

1

, . . . , v

p

}

,

{e

1

, . . . , e

q

}

,

{f

1

, . . . f

r

}

the sets ofverti es,edges, fa esof

T

Θ

, respe tively. We re all that

ϕ

maps

k

-fa es of

H

to

k

-fa es of

T

Θ

. We saythat

H

is

Θ

-uniform iffor all

k

_{∈ {0, 1, 2}}

,all

k

-fa esof

H

mappedtoa

k

-fa eof

T

Θ

havethesamevalen y. Toputitanother way,a

Θ

- onservativehypermap

H

is

Θ

-uniformifforall

k

∈ {0, 1, 2}

,

k

-fa es ontainingags in the same

Θ

-orbithave the same valen y. When

H

isa

Θ

-uniform hypermap su hthat all verti es of

H

mapped to the vertex

v

i

of

T

Θ

have valen y

l

i

, all edges of

H

mapped to the edge

e

j

of

T

Θ

have valen y

m

j

and all fa es of

H

mapped to the fa e

f

k

of

T

Θ

have valen y

n

k

, we saythat

H

has

Θ

-type

(l

1

, . . . , l

p

; m

1

, . . . m

q

; n

1

, . . . n

r

)

. We mayassume, withoutloss of generality, that

l

1

≤ · · · ≤ l

p

,

m

1

≤ · · · ≤ m

q

and

n

1

≤ · · · ≤ n

r

. A hypermap is alled bipartite-uniform ifit is

∆

ˆ

0

-uniform. The bipartite-type of a bipartite-uniform hypermap

B

is its

∆

ˆ

0

-type

(l

1

, l

2

; m; n)

, where

l

1

and

l

2

are the valen ies (not ne essarily distin t)of the verti es of

B

, and

m

and

n

are the valen ies of the edges and the fa es of

B

. Sin e

B

is bipartite-un iform,

B

is bipartite and, by Lemma 1.3.6,

m

and

n

are even. Moreover, a

∆

ˆ

k

-uniformhypermapis alled

k

-bipartite-uniform;wealsousethetermsvertex-bipartite -uniform, edge-bipartite-uniform and fa e-bipartite-uniform instead of

0

-bipartite-uni form,

1

-bipartite-uniform and

2

-bipartite-uniform, respe tively.

Lemma 1.3.9. Let

Θ

be a normal subgroup of

∆

and

H

a

Θ

- onservative hypermap. 1. If

H

is

Θ

-regular, then

H

is

Θ

-uniform.

2. If

Θ

is

∆

+

,

∆

0

,

∆

1

,

∆

2

or

∆

012

, then

H

is

Θ

-uniformif and onlyif

H

is uniform. 3. If

Θ

is

∆

+0ˆ

0

,then

H

is

∆

+0ˆ

0

-uniform ifand onlyif

H

isbipartite-uniform.

Proof. 1. Let

k

∈ {0, 1, 2}

,

ω

∈ Ω

H

,

H = Stab

∆

(ω)

and

g

∈ Θ

. If

H

is

Θ

-regular, then

H ⊳ Θ

andhen e

H

g

_{= H}

. In parti ular,the

k

-fa es ontaining

ω

and

ωg

have the same valen y. 2. and 3. One an easily see that the hypermaps

T

Θ

, where

Θ

is

∆

+

,

∆

0

,

∆

1

or

∆

2

, have 1 vertex,1edgeand1fa e;the hypermaps

T

∆

ˆ

0

and

T

∆

+0ˆ

0

have2verti es,1edgeand1fa e. A uniform hypermap is

k

-bipartite-uniform if and only if it is

k

-bipartite. Examples of

Θ

-uniform hypermapsthatarenot

Θ

-regular an befound in Chapter4.

1.4 The Euler formula and the Hurwitz bound

A theorem of Hurwitz [38℄ ( f. [27, 18, 61℄) states that an upper bound for the number of onformalautomorphisms ofa ompa t Riemann surfa ewith genus

g

greater than one(that is, homeomorphisms of the surfa e onto itself preserving the lo al stru ture) is

84(g

− 1)

. It has been proved by Jones and Singerman [49℄ that the group of orientation- preserving automorphisms of a map

M

on an orientable surfa e of genus

g

is isomorphi to a group of onformal automorphisms of a ompa t Riemann surfa e with the same genus, and hen e bounded by

84(g

− 1)

. Moreover, the number of automorphism of a map

M

is bounded by

168(g

_{− 1)}

, if

M

is orientable, and by

84(g

− 2)

, otherwise (see, for instan e, Theorem 4.2.2 of[61℄).

Ouraiminthisse tionistopresentmethodsforndingallpossibletypes(resp. bipartite-types)of uniform (resp. bipartite-un iform) hypermaps ona given surfa e. Wegivea relation betweenthe Euler hara teristi ,number ofagsandtype(resp. bipartite-type)of auniform (resp. bipartite-uniform)hypermap,andthenweuseittondboundsforthenumbersofags ofuniform (resp. bipartite-un i form) hypermapswith agiven negative Euler hara teristi .

Using the well-knownEuler (polyhedral)formulaone aneasily getthe following result.

Lemma1.4.1(Eulerformulaforhypermaps) . Let

H

bea hypermapwith

V

verti es,

E

edges,

F

fa es and Euler hara teristi

χ

. Then

χ = V + E + F

₋

|Ω

H

|

2

.

(1.2)

When

H

is uniform of type

(l, m, n)

,

V =

|Ω

H

|/2l

,

E =

|Ω

H

|/2m

and

F =

|Ω

H

|/2n

. Repla ingthe valuesof

V

,

E

and

F

in formula(1.2),weget:

Corollary 1.4.2 (Euler formula for uniform hypermaps) . Let

H

be a uniform hypermap of type

(l, m, n)

with Euler hara teristi

χ

. Then

χ =

|Ω

H

|

2



1

l

+

1

m

+

1

n

− 1



.

(1.3)

When

H

isbipartite-uniformofbipartite-type

(l

1

, l

2

; m; n)

, ea h

∆

ˆ

0

-orbithas

|Ω

H

|/2

ags, and so the numbers of verti es in the

∆

ˆ

0

-orbits are

|Ω

H

|/4l

1

and

|Ω

H

|/4l

2

. Then

H

has

V =

|Ω

H

|/4l

1

+

|Ω

H

|/4l

2

verti es,

E =

|Ω

H

|/2m

edgesand

F =

|Ω

H

|/2n

fa es. Repla ingthe values of

V

,

E

and

F

in formula(1.2),we get:

Corollary 1.4.3 (Euler formula for bipartite-uniform hypermaps) . Let

H

be a bipartite-uniform hypermap of bipartite-type

(l

1

, l

2

; m; n)

withEuler hara teristi

χ

. Then

χ =

|Ω

H

|

2



1

2l

1

+

1

2l

2

+

1

m

+

1

n

− 1



.

(1.4)

Lemma 1.4.4. If

H

isa hypermap su hthat all verti es have valen y1, then

H

isa uniform hypermap on the sphere of type

(1, k, k)

, where

k

is the number of verti es. Furthermore,

H

is regular.

Proof. If all verti es have valen y1, then

R

1

R

2

∈ H

g

, for all

g

∈ ∆

, so

R

1

R

2

∈ H

∆

. Conse-quently,

H

∆

R

1

= H

∆

R

2

and

Mon(

_{H) = ∆/H}

∆

=

hH

∆

R

0

, H

∆

R

1

, H

∆

R

2

i = hH

∆

R

2

, H

∆

R

0

i = hH

∆

R

0

, H

∆

R

1

i.

(1.5) Sin e

Mon(

H)

a ts transitively on

Ω

H

,

H

has exa tly one

hH

∆

R

2

, H

∆

R

0

i

-orbit and one

hH

∆

R

0

, H

∆

R

1

i

-orbit, that is, 1 edge and 1 fa e, both with valen ies

k :=

|Ω

H

|/2

. Obvi-ously,

H

is uniform of type

(1, k, k)

and has

k

verti es, 1 edge and 1 fa e. Finally, using the Euler formula for hypermaps (Lemma 1.4.1), we see that

χ

H

= V + E + F

− |Ω

H

|/2 =

|Ω

H

|/2 + 1 + 1 − |Ω

H

|/2 = 2

.

Now assume that

H

is a uniform hypermap of type

(l, m, n)

. By Corollary 1.4.2,

H

is imbedded on a surfa e with Euler hara teristi greater than, equal to, or smaller than 0 dependingonwhether

1/l+1/m+1/n

isgreaterthan,equalto,orsmallerthan1,respe tively. Lemma 1.4.5. Let

l

,

m

,

n

be positive integers su h that

l

≤ m ≤ n

, and

S =

1

l

+

m

1

+

1

n

. Then 1.

S > 1

if andonly if

(l, m, n)

is

(1, j, k)

,

(2, 2, k)

,

(2, 3, 3)

,

(2, 3, 4)

or

(2, 3, 5)

,

j, k

∈ N

; 2.

S = 1

if andonly if

(l, m, n)

is

(2, 3, 6)

,

(2, 4, 4)

or

(3, 3, 3)

; 3.

S < 1

if andonly if

S

≤

1

2

+

1

3

+

1

7

=

41

42

.

Proof. 1. When

S > 1

,

3/l

≥ S > 1

, and so

l < 3

. If

l = 1

, then

S > 1

; else, if

l = 2

, then

2/m

≥ 1/m + 1/n > 1/2

andhen e

m < 4

. Then

m = 2

,or

m = 3

and

n < 6

.

2. When

S = 1

,

3/l

≥ S = 1 > 1/l

, and so

1 < l

≤ 3

. If

l = 2

, then

2/m

≥ 1/m + 1/n =

1/2 > 1/m

, so

2 < m

≤ 4

and

n = 2m/(m

− 2)

. Then

m = 3

and

n = 6

, or

m = n = 4

. If

l = 3

, then

1 = 3/l

≥ S = 1

implies that

l = m = n = 3

.

3. Assume that

l, m, n

arepositive integerssu h that

l

≤ m ≤ n

and

S < 1

. Then: (a) if

l = 2

,

m = 3

and

n > 6

, then

S

≤

1

2

+

1

3

+

1

7

=

41

42

; (b) if

l = 2

,

m = 4

and

n > 4

, then

S

≤

1

2

+

1

4

+

1

5

=

19

20

; ( ) if

l = 2

and

m > 4

, then

S

≤

1

2

+

1

5

+

1

5

=

10

9

; (d) if

l = 3

and

n > 3

, then

S

≤

1

3

+

1

3

+

1

4

=

11

12

; (e) if

l > 3

, then

S

≤

1

4

+

1

4

+

1

4

=

3

4

.

Using Corollary 1.4.2together with Lemma 1.4.5,we getthe following well-knownresult.

Theorem 1.4.6(Hurwitzboundfor uniform hypermapswith negativeEuler hara teristi ) . If

H

is a uniform hypermap withnegativeEuler hara teristi

χ

,then

|Ω

H

| ≤ −84χ

.

Now we determine bounds for the number of ags of a bipartite-un iform hypermap with givennegativeEuler hara teristi .

Let

B

be a bipartite-un iform hypermap of type

(l

1

, l

2

; m; n)

. A ording to Lemma 1.3.6,

m

and

n

areeven. Let

(a, b, c, d) = (l

1

, l

2

, m/2, n/2)

. By Corollary 1.4.3,

H

is imbedded ona surfa ewith Euler hara teristi

> 0

,

= 0

or

< 0

dependingonwhether

1/a + 1/b + 1/c + 1/d

isgreater than, equal to,or smallerthan 2,respe tively.

Lemma 1.4.7. Let

a

,

b

,

c

and

d

be positive integers su h that

a

≤ b ≤ c ≤ d

, and

T =

1

a

+

1

b

+

1

c

+

1

d

. Then:

1.

T > 2

ifandonlyif

(a, b, c, d)

is

(1, 1, j, k)

,

(1, 2, 2, k)

,

(1, 2, 3, 3)

,

(1, 2, 3, 4)

or

(1, 2, 3, 5)

, where

j, k

∈ N

;

2.

T = 2

if and onlyif

(a, b, c, d)

is

(1, 2, 3, 6)

,

(1, 2, 4, 4)

,

(1, 3, 3, 3)

or

(2, 2, 2, 2)

; 3.

T < 2

if and onlyif

T

≤

1

1

+

1

2

+

1

3

+

1

7

=

83

42

. Proof. Let

S =

1

b

+

1

c

+

1

d

. Then:

(a)if

a = 1

,then

T > 2

,

= 2

or

< 2

ifand onlyif

S > 1

,

= 1

or

< 1

, respe tively; (b)if

a = b = c = d = 2

,then

T = 2

; ( )if

a = 2

and

d > 2

, then

T

≤

1

2

+

1

2

+

1

2

+

1

3

=

11

6

; (d)if

a > 2

, then

T

≤

1

3

+

1

3

+

1

3

+

1

3

=

4

3

. Nowthe resultfollows fromLemma 1.4.5.

Finally, usingCorollary 1.4.3 together with Lemma 1.4.7, we get:

Theorem 1.4.8 (Hurwitz bound for bipartite-uniform hypermapswith negative Euler har-a teristi ). If

H

is a bipartite-uniform hypermap with negative Euler hara teristi

χ

, then

|Ω

H

| ≤ −168χ

.

1.5 Duality

Everyautomorphism

θ

of

∆

givesrisetoanoperationonhypermapsbytransforminga hyper-map

H

with hypermap subgroup

H

, to its operation-dual,

D

θ

(

H)

, with hypermap subgroup

Hθ

(see [41, 43, 44℄ for more details),thatis, if

H = (∆/

r

H, H

∆

R

0

, H

∆

R

1

, H

∆

R

2

)

, then

D

θ

(

H) = (∆/

r

Hθ, (Hθ)

∆

R

0

, (Hθ)

∆

R

1

, (Hθ)

∆

R

2

)

= (∆/

r

Hθ, H

∆

θR

0

, H

∆

θR

1

, H

∆

θR

2

).

When

θ

isan innerautomorphism,

H

and

Hθ

are onjugate in

∆

and,byCorollary 1.3.2,

H

and

D

θ

(

H)

are isomorphi . Ea h permutation

σ

∈ S

{0,1,2}

indu es an outer automorphism (that is, a non-inner automorphism)

σ : ∆

→ ∆

su h that

R

i

σ = R

iσ

, for all

i = 0, 1, 2

. By abuse of language, we speak of

D

σ

, meaning the operator

D

σ

. These operations, presented by Ma hì in [52℄, transform one hypermap

H

to another byrenaming its verti es, edges and fa es. To be more pre ise, the

k

-fa e of

H

ontaining the ag

Hg

orrespondsto the

kσ

-fa e of

D

σ

(

H)

ontaining

Hσgσ

. In parti ular, they have the same valen y. James [41℄ showed that the operations on hypermaps form an innite group, Out

(∆)

, isomorphi to

P GL

2

(Z)

ontaining Ma hì's operations.

Lemma 1.5.1. Let

σ

∈ S

{0,1,2}

and

σ : ∆

→ ∆

dened as above. Then

∆

+

_{σ = ∆}

+

,

∆

ˆ

k

σ = ∆

kσ

and

∆

k

_{σ = ∆}

kσ

, for all

k

∈ {0, 1, 2}

.

Proposition 1.5.2(Propertie s of

D

σ

) . Let

H

,

G

be hypermaps and

σ, τ

∈ S

{0,1,2}

. Then: 1.

D

1

(

H) = H

;

D

τ

(D

σ

(

H)) = D

στ

(

H)

;

2.

H → G

if and onlyif

D

σ

(

H) → D

σ

(

G)

;

H ∼

=

G

if and onlyif

D

σ

(

H) ∼

= D

σ

(

G)

; 3.

H

is

Θ

- onservative if andonly if

D

σ

(

H)

is

Θσ

- onservative;

4.

H

is

Θ

-uniform if andonly if

D

σ

(

H)

is

Θσ

-uniform; 5.

H

is

Θ

-regular if andonly if

D

σ

(

H)

is

Θσ

-regular; 6.

H

and

D

σ

(

H)

have thesame underlying surfa e; 7.

Aut(

H) ∼

= Aut(D

σ

(

H))

and

Mon(

H) ∼

= Mon(D

σ

(

H))

.

Asan immediate orollaryto Proposition 1.5.2 we get

Corollary 1.5.3. 1.

H

is uniform (resp.

k

-bipartite-uniform) if and only if

D

σ

(

H)

is uniform (resp.

kσ

-bipartite-uniform);

2.

H

is regular (resp. orientably-regular,

k

-pseudo-orientably-regul a r ,

k

-bipartite-regul a r) if and only if

D

σ

(

H)

is regular (resp. orientably-regula r,

kσ

-pseudo-orie ntabl y-regular,

kσ

-bipartite-regul a r);

3. Every

k

-pseudo-orientabl y-regul a r hypermap isuniform.

This result shows that, up to duality, a 2-restri tedly-reg ular hypermap is orientably- hiral, pseudo-orientably- hir al or bipartite- h ir al. Consequently, the lassi ation of all 2-restri tedly-regul ar hypermaps on a surfa e

S

an be derived from the lassi ation of these 3 typesof hypermapson

S

.

The 2-skeleton of a onvex polyhedron in

R

3

an be viewed asa map on the sphere. In parti ular, the Platoni solids give rise to 5 regular maps on the sphere. For simpli ity, we will not dierentiate thesemaps from the orrespondingPlatoni solids. We denoteby

T

,

C

,

O

,

D

and

I

the tetrahedron, the ube (or hexahedron), the o tahedron, the dode ahedron and the i osahedron. These maps have type

(3, 2, 3)

,

(3, 2, 4)

,

(4, 2, 3)

,

(3, 2, 5)

and

(5, 2, 3)

, respe tively. It is well-known that if

H

is one of these hypermaps and

(l, m, n)

is the type of

H

, then

H

has hypermap subgroup

h(R

1

R

2

)

l

_{, (R}

2

R

0

)

m

, (R

0

R

1

)

n

i

∆

, automorphism group

Aut(

_{H) ∼}

= ∆(l, m, n)

, and that

T ∼

= D

(02)

(

T )

,

O ∼

= D

(02)

(

C)

and

I ∼

= D

(02)

(

D)

. For more information onthese hypermaps,see Se tion2.1.

Given

k

∈ N

,thedihedral hypermapoforder

k

,

D

k

,andthepolygonoforder

k

,

P

k

,arethe regular hypermapson the sphere of type

(k, k, 1)

and

(2, 2, k)

, and with hypermap subgroup

h(R

1

R

2

)

k

, (R

2

R

0

)

k

, R

0

R

1

i

∆

and

h(R

1

R

2

)

2

_{, (R}

2

R

0

)

2

, (R

0

R

1

)

k

i

∆

, respe tively. In Figure 1.1 we display

D

8

and

P

4

. The star hypermap of order

k

is the hypermap

S

k

= D

(02)

(

D

k

)

. The dihedral hypermap of order

k

has

2k

ags, 1vertex,1 edge and

k

fa es;the polygon of order

k

has

4k

ags,

k

verti es,

k

edges and2fa es. Using Corollary1.4.2 we anseethatboth

D

k

and

P

k

areon the sphere. In[15℄, Breda and Jonesdenoted the hypermaps

P

k

(with

k

odd) and

D

(01)

(

D

k

)

by

D

⊖

k

and

D

∗