Department of Mathematics

University of Fribourg (Switzerland)

On the existence and growth properties

of hyperbolic Coxeter groups

THESIS

presented to the Faculty of Science and Medicine of the University of

Fribourg (Switzerland)

in consideration for the award of the academic grade of

Doctor of Philosophy in Mathematics

Naomi Bredon

from

France

Thesis No: 8046

Uniprint Fribourg

2024

Accepted by the Faculty of Science and Medicine of the University of Fri-

bourg (Switzerland) upon the recommendation of the jury:

Prof. Dr. Anand Dessai

University of Fribourg (Switzerland), President of the jury

Prof. Dr. Ruth Kellerhals

University of Fribourg (Switzerland), Thesis supervisor

Prof. Dr. Yohei Komori

Waseda University (Japan), External expert

Dr. Jean Raimbault

University of Aix- Marseille (France), External expert

Prof. Dr. Pavel Tumarkin

Durham University (United Kingdom), External expert

Fribourg, 12.09.2024

Thesis supervisor

Prof. Dr. Ruth Kellerhals

Dean

Prof. Dr. Ulrich Ultes-Nitsche

Abstract

A Coxeter polyhedron in Hnis a convex polyhedron of ﬁnite volume all of

whose dihedral angles are integral submultiples of π. The group generated

by the reﬂections in the facets of a hyperbolic Coxeter polyhedron is called

a hyperbolic Coxeter group. While hyperbolic Coxeter polyhedra exist only

for n≤995, they are far from being classiﬁed for dimensions beyond 3.

In this thesis, we study hyperbolic Coxeter polyhedra and their reﬂection

groups from two diﬀerent point of views.

In the ﬁrst part of our work, we consider Coxeter polyhedra with mutually

intersecting facets all of whose dihedral angles are π

2,π

3and (at least one) π

Since their associated hyperbolic Coxeter groups satisfy a crystallographic

condition, we call them ADEG-polyhedra. Our ﬁrst main result provides the

classiﬁcation of all ADEG-polyhedra in Hn. We discover a new polyhedron

P⋆in H9, and present various of its properties. Furthermore, for n≥7, we

show that Coxeter polyhedra in Hnwith mutually intersecting facets have

dihedral angles of the form π

mwith m≤6, only.

In the second part of our work, we study growth minimality properties of

Coxeter groups acting on Hnfor dimensions n≥4. To this end, we es-

tablish a new method to identify the groups realizing smallest growth rate

and exploit it subsequently by distinguishing between the cocompact and the

non-cocompact cases. One of our results concerns the cocompact case and

dimensions n= 4 and 5 while the other result describes the non-cocompact

case and dimensions 4 ≤n≤9. In both settings, we are able to identify

the Coxeter groups of minimal growth rate. It turns out that they are all

intimately related to the fundamental groups of (compact arithmetic resp.

cusped) hyperbolic n-orbifolds of minimal volume.

R´

esum´

Un poly`edre de Coxeter dans Hnest un poly`edre convexe de volume ﬁni

dont tous les angles di`edres sont des sous-multiples entiers de π. Le groupe

engendr´e par les r´eﬂexions par rapport aux facettes d’un poly`edre de Coxeter

hyperbolique est appel´e un groupe de Coxeter hyperbolique. Bien qu’il soit

´etabli que les poly`edres de Coxeter hyperboliques existent seulement pour

n≤995, ils sont loin d’ˆetre classiﬁ´es au del`a de la dimension 3.

Cette th`ese porte sur l’´etude des poly`edres de Coxeter hyperboliques et leurs

groupes de r´eﬂexions selon deux points de vue diﬀ´erents.

Dans la premi`ere partie de notre travail, nous consid´erons des poly`edres de

Coxeter hyperboliques dont toutes les facettes s’intersectent, et ce, en des

angles di`edres π

2,π

3et (au moins un) π

6. Les groupes de Coxeter associ´es `a de

tels poly`edres satisfont une condition crystallographique, et nous les bapti-

sons poly`edres ADEG. Notre premier r´esultat principal ´etablit la classiﬁca-

tion compl`ete des poly`edres ADEG. Nous d´ecouvrons un nouveau poly`edre

P⋆dans H9, et pr´esentons plusieurs de ses propri´et´es. De plus, nous mon-

trons que, pour n≥7, les poly`edres de Coxeter dont toutes les facettes

s’intersectent ont des angles di`edres de la forme π

mpour m≤6 seulement.

Dans la seconde partie de notre travail, nous ´etudions la propri´et´e de crois-

sance minimale des groupes de Coxeter de covolume ﬁni agissant sur Hn

pour n≥4. Pour cela, nous ´etablissons une nouvelle m´ethode pour identiﬁer

les groupes dont le taux de croissance est minimal. Nous exploitons cette

m´ethode en distinguant les cas cocompact et non-cocompact.

Un de nos r´esultats concerne le cas cocompact en dimension n= 4 et 5, tandis

que notre autre r´esultat porte sur le cas non-cocompact pour 4 ≤n≤9.

Dans les deux cas, nous identiﬁons les groupes de Coxeter de plus petit

taux de croissance. Ces derniers sont en fait ´etroitement li´es aux groupes

fondamentaux des n-orbivari´et´es (compactes arithm´etiques, respectivement

cuspid´ees) hyperboliques de volume minimal.

Zusammenfassung

Ein Coxeter-Polyeder in Hnist ein konvexes Polyeder von endlichem Vol-

umen, dessen Diederwinkel ganzzahlige Teiler von πsind. Die Gruppe,

die durch die Spiegelungen in den Facetten eines hyperbolischen Coxeter-

Polyeders erzeugt wird, heisst eine hyperbolische Coxeter-Gruppe. W¨ahrend

hyperbolische Coxeter-Polyeder nur f¨ur n≤995 existieren, ist man dennoch

weit entfernt von deren Klassiﬁkation jenseits der Dimension drei.

In dieser Arbeit untersuchen wir hyperbolische Coxeter-Polyeder und deren

Spiegelungsgruppen aus zwei verschiedenen Blickwinkeln. Im ersten Teil un-

serer Arbeit betrachten wir Coxeter-Polyeder mit sich gegenseitig schneiden-

den Facetten, deren Diederwinkel alle gleich π

2,π

3und (mindestens einem)

6sind. Da ihre zugeh¨origen hyperbolischen Coxeter-Gruppen eine kristallo-

graphische Bedingung erf¨ullen, nennen wir sie ADEG-Polyeder. Unser erstes

Hauptergebnis liefert die Klassiﬁkation aller ADEG-Polyeder in Hn. Dabei

entdecken wir ein neues Polyeder P⋆in H9und pr¨asentieren verschiedene

seiner Eigenschaften. Ausserdem zeigen wir f¨ur n≥7, dass Coxeter-Polyeder

in Hnmit sich gegenseitig schneidenden Facetten nur Diederwinkel der Form

mmit m≤6 haben k¨onnen.

Im zweiten Teil unserer Arbeit untersuchen wir die Eigenschaft des mini-

malem Wachstums f¨ur Coxeter-Gruppen, die auf Hnf¨ur Dimensionen n≥4

operieren. Zu diesem Zweck etablieren wir eine neue Methode, um die Grup-

pen mit minimaler Wachstumsrate zu identiﬁzieren, und wenden diese an-

schliessend an, indem wir zwischen dem kokompakten und dem nicht kokom-

pakten Fall unterscheiden.

Eines unserer beiden Ergebnisse betriﬀt den kokompakten Fall und die Di-

mensionen n = 4 und 5, w¨ahrend das andere Ergebnis den nicht-kokompakten

Fall und die Dimensionen 4 ≤n≤9 behandelt. In beiden F¨allen sind wir

in der Lage, die Coxeter-Gruppen mit minimaler Wachstumsrate zu identi-

ﬁzieren. Es stellt sich heraus, dass sie alle eng mit den Fundamentalgruppen

von (kompakten arithmetischen bzw. gespitzten) hyperbolischen n-Orbifolds

von minimalem Volumen verkn¨upft sind.

iii

Remerciements

Je tiens `a exprimer ma profonde gratitude envers Ruth Kellerhals pour son

soutien et ses conseils avis´es tout au long de l’´ecriture de ce manuscrit. Ce fˆut

un honneur de travailler avec elle, et je garderai en m´emoire nos nombreux

´echanges stimulants, nos excursions math´ematiques, et tout ce que j’ai appris

`a ses cˆot´es. Son exp´erience a grandement contribu´e au d´eveloppement de mes

comp´etences ces derni`eres ann´ees, sur les plans acad´emique, professionnel, et

mˆeme personnel. Pour tout cela, je lui suis inﬁniment reconnaissante.

Je remercie les membres du jury Yohei Komori, Jean Raimbault et Pavel

Tumarkin d’avoir accept´e de constituer le jury et pour leur soutien dans mon

projet. J’aimerais aussi remercier Anand Dessai qui pr´eside le jury.

Ma gratitude s’´etend ´egalement envers ma famille pour leur ´ecoute et leurs

nombreux encouragements. Merci `a vous mes chers parents, mes ch`eres sœurs

et mon cher fr`ere, pour votre soutien in´ebranlable. Plus largement, je remer-

cie mes amis, d’ici et d’ailleurs, et tous ceux qui m’ont accompagn´ee dans

cette aventure.

Quant `a chacun des membres du d´epartement, et `a ceux d’autres instituts,

qui ont crois´e ma route, merci pour ces moments de convivialit´e. Finalement,

merci `a mes ´etudiants pour leur dynamisme et leur investissement, vos ques-

tionnements m’ont moi-mˆeme souvent enrichie. Vous tous avez contribu´e `a

rendre chaque ´etape de ce voyage m´emorable et signiﬁcative.

Merci `a chacun d’entre vous

Table of Contents

Introduction 1

I Preliminaries 5

1 Coxeter groups and Coxeter polyhedra 6

1 Geometric spaces of constant curvature . . . . . . . . . . . . . 6

2 Hyperplanes, polyhedra and Gram matrix . . . . . . . . . . . 9

3 Abstract and geometric Coxeter groups . . . . . . . . . . . . . 14

3.1 The Coxeter diagram of a Coxeter group . . . . . . . . 16

3.2 Spherical and Euclidean Coxeter groups . . . . . . . . 17

3.3 Hyperbolic Coxeter groups . . . . . . . . . . . . . . . . 18

2 Root systems and fundamental weights 24

1 Root systems and Weyl groups . . . . . . . . . . . . . . . . . 24

2 Dynkin diagrams and extended Dynkin diagrams . . . . . . . 27

3 Classiﬁcation of root systems of type A,D,Eand G2..... 27

II On the existence of hyperbolic Coxeter groups 32

3 Some classification results 33

1 Inlowdimensions......................... 34

2 In higher dimensions . . . . . . . . . . . . . . . . . . . . . . . 37

2.1 Coxeter polyhedra with a small number of facets . . . . 37

2.2 Coxeter polyhedra with mutually intersecting facets . . 42

4 Coxeter polyhedra with dihedral angles π

2,π

3and π

645

1 The general strategy . . . . . . . . . . . . . . . . . . . . . . . 48

2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . 49

2.1 Firststeps......................... 49

2.2 Existence of a component e

G2.............. 50

2.3 From a G2-face to an admissible set of vectors . . . . . 50

2.4 The classiﬁcation of ADEG-polyhedra . . . . . . . . . 55

3 Further results and comments . . . . . . . . . . . . . . . . . . 69

3.1 Properties of the polyhedron P⋆............. 69

3.2 An angular obstruction . . . . . . . . . . . . . . . . . . 73

III On the growth rates of hyperbolic Coxeter groups 74

5 The growth rates of Coxeter groups 75

1 Growth series and growth rates . . . . . . . . . . . . . . . . . 75

2 About the arithmetic nature of growth rates . . . . . . . . . . 80

3 Comparing growth rates . . . . . . . . . . . . . . . . . . . . . 82

3.1 Partial order and growth monotonicity . . . . . . . . . 82

3.2 Ausefullemma...................... 85

6 Minimal growth rates for hyperbolic Coxeter groups 87

1 Inlowdimensions......................... 88

2 In higher dimensions . . . . . . . . . . . . . . . . . . . . . . . 89

3 Proofs of the two main results . . . . . . . . . . . . . . . . . . 90

3.1 The cocompact case . . . . . . . . . . . . . . . . . . . 91

3.2 The non-cocompact case . . . . . . . . . . . . . . . . . 92

IV Appendices 95

A Data for Prokhorov’s formula 96

B Admissible pairs 98

C Normal vectors and Vinberg form 114

D The two reprints 119

Bibliography 148

Introduction

Let Hnbe the hyperbolic space of dimension n≥2. A Coxeter polyhedron

P⊂Hnis a convex polyhedron of ﬁnite volume all of whose dihedral angles

are integer submultiples of π. Associated with Pis the discrete group Γ(P)

in IsomHngenerated by the reﬂections in the facets of P. Such groups are

called hyperbolic Coxeter groups and form a natural but important family

of transformation groups arising in several diﬀerent contexts, ranging from

algebra, geometry to topology. However, in contrast to the Euclidean and

the spherical cases, hyperbolic Coxeter polyhedra are far from being entirely

understood.

Our thesis is devoted to this theme and consists of three parts. While Part

I contains the preparatory material, our two key achievements are presented

in Part II and in Part III.

Firstly, we consider a class of Coxeter polyhedra in Hnwith prescribed com-

binatorial and angular structure. In fact, instead of ﬁxing the number of

their facets in terms of the dimension n, as happened often in works of other

authors, we suppose that the polyhedra have mutually intersecting facets

and dihedral angles given by π

2,π

3and (at least one) π

6. Motivated by work of

Prokhorov, and in view of their relation to crystallographic Coxeter groups

of type A, D, E and G2, we call them ADEG-polyhedra. Our central result

can be stated as follows.

Theorem. Let P⊂Hnbe an ADEG-polyhedron. Then, Pis one of the 24

Coxeter polyhedra depicted in Table 4.0.1. In particular, Pis non-compact

for n > 2, non-simple for n > 3, and Pis of dimension n≤11. Furthermore,

Pis combinatorially equal to one of the following polyhedra.

✧Pis a triangle.

✧Pis a tetrahedron.

✧Pis a doubly-truncated 5-simplex.

✧Pis a pyramid over a product of two or three simplices.

✧Pis the polyhedron P⋆⊂H9with 14 facets depicted as follows.

The polyhedron P⋆⊂H9is a new discovery, and it is given here by its Coxeter

diagram. Such a diagram can be read as follows. The nodes correspond to

facets, and they are joined by a simple edge or an edge with label m≥4 if the

facets are not orthogonal but intersect under the angle π

3or π

m, respectively.

The proof of the classiﬁcation theorem above is in parts based on Prokhorov’s

work in the ADE-case and given in Chapter 4. Of fundamental importance to

us are the results of Felikson and Tumarkin characterizing the combinatorics

of Coxeter polyhedra with mutually intersecting facets, as well as Borcherds’

theorem helping us to identify faces of codimension two as Coxeter polyhedra.

At the end of Part II, we present a few properties of P⋆and of its associated

Coxeter group Γ⋆= Γ(P⋆). We show that its volume is a rational multiple q

of ζ(5)/22,295,347,200 with q∈Q>1. Moreover, the group Γ⋆is arithmetic

(over Q), and its commensurability class contains all Coxeter simplex groups

and all Coxeter pyramid groups as well as the group related to Prokhorov’s

ADE-polyhedron.

Let us add that our proof methods are general and can be applied to Coxeter

polyhedra with mutually intersecting facets and for any set of dihedral angles.

However, as we proved in Proposition 4.3.3, the dihedral angles of Coxeter

polyhedra with mutually intersecting facets of dimensions beyond 6 must

be of the form π

mwith m≤6. Although a complete classiﬁcation of these

polyhedra can therefore be realized in ﬁnite time and for all dimensions, we

did not pursue this direction of research.

Secondly, we present in Part III our results about hyperbolic Coxeter groups

of minimal growth rate in dimensions beyond three. Since these results have

already been published at an earlier stage of this thesis work, we are content

to summarize them in a condensed but nevertheless complete way. The ﬁrst

result1in this context is given by Theorem 6.2.1 and identiﬁes the cocompact

hyperbolic Coxeter groups of minimal growth rate in dimensions n= 4 and 5.

More precisely, for dimension 4, it is the group Γc

4associated with the compact

simplex with Coxeter symbol [5,3,3,3] while for dimension 5, it is the group

Γc

5associated with the compact prism with Coxeter symbol [5,3,3,3,3,∞].

The second result2, given by Theorem 6.2.2, identiﬁes the non-cocompact

hyperbolic Coxeter groups of minimal growth rate in dimensions 4 ≤n≤9.

The minimizing groups Γnare depicted below.

Γ4

4Γ5

Γ6

4Γ7

Γ8Γ9

In both cases, the minimizers are unique and closely related to the funda-

mental groups of compact arithmetic and cusped hyperbolic n-orbifolds of

minimal volume, respectively.

The work ﬁnishes with several appendices. Appendix A and Appendix B

display some material and lists in connection with the proof of the above

Theorem. Appendix C contains the technical details relevant for the proof

of Proposition 4.3.1 about the commensurability class of Γ⋆. Lastly, the

two articles mentioned about constitute Appendix D. Let us indicate that

we contributed signiﬁcantly to the research in our joint work with Ruth

Kellerhals.

1N. Bredon, R. Kellerhals, Hyperbolic Coxeter groups and minimal growth rates in

dimensions four and ﬁve, Groups Geom. Dynamics 16 (2022), 725–741.

2N. Bredon, Hyperbolic Coxeter groups of minimal growth rates in higher dimensions,

Canad. Math. Bull. 66 (2023), 232–242.

In our times, geometers are still exploring

those new Wonderlands, partly for the

sake of their applications to cosmology

and other branches of science, but much

more for the sheer joy of passing through

the looking glass into a land where

the familiar lines, planes, triangles,

circles and spheres are seen to behave

in strange but precisely determined ways.

H. S. M. Coxeter

Part I:

Preliminaries

CHAPTER 1

Coxeter groups and Coxeter polyhedra

Let Xnbe a geometric space of dimension n≥2 and of constant sectional

curvature 0,1 or −1. Let IsomXnbe the isometry group of Xn. We are

interested in discrete groups in IsomHngenerated by ﬁnitely many reﬂections

with respect to hyperplanes of Xn.

For their characterization, we ﬁrst present vector models for the spaces Xn.

Our interest lies especially in the case Xn=Hn. After that, we describe

hyperplanes and polyhedra in Xn.Then, we introduce the important notion

of Coxeter groups. We ﬁrst consider them as abstract objects, and then rep-

resent some of them as geometric Coxeter groups, that is, Coxeter groups

realized as discrete subgroups of IsomXngenerated by reﬂections in hyper-

planes in Xn. We characterize geometric Coxeter groups in terms of their

Coxeter diagrams and by means of the Gram matrix of their fundamental

polyhedra.

As general references for this chapter, we quote [16, 37, 82, 83, 84].

1 Geometric spaces of constant curvature

Let n≥2. The only simply connected complete Riemannian manifolds of

constant sectional curvature of dimension nare, up to isometry, the Euclidean

space En, the sphere Snand the hyperbolic space Hn. In what follows, we

consider each of these spaces as a metric space.

Denote by K∈ {0,1,−1}the constant sectional curvature of the space Xn.

1. Geometric spaces of constant curvature

For x= (x1, x2, . . . , xn+1) and y= (y1, y2, . . . , yn+1)∈Rn+1, deﬁne the

bilinear form

⟨x, y⟩K:=

i=1

xiyi+Kxn+1yn+1 .

We denote by || · ||Kthe associated (pseudo-)norm. If the context is clear,

we simply write ⟨·,·⟩ := ⟨·,·⟩Kand || · || := || · ||K.

The bilinear form ⟨., .⟩0is positive semideﬁnite, and the bilinear form ⟨., .⟩1

is positive deﬁnite. The bilinear form ⟨., .⟩−1is indeﬁnite of signature (n, 1),

and it is called the Lorentzian form.

The spaces Xnadmit the following linear models.

(i) We identify the space Enwith the hyperplane

En={x∈Rn+1 |xn+1 = 0}

in Rn+1. It is naturally endowed with the metric dE(x, y) = p⟨x−y, x −y⟩0

for any x, y ∈En.

(ii) The space Snis given by

Sn={x∈Rn+1 | ⟨x, x⟩1= 1},

together with the usual angular metric dS(x, y) = arccos⟨x, y⟩1for x, y ∈Sn.

(iii) Denote by Rn,1the Lorentz-Minkowski space, that is, the space Rn+1

endowed with the Lorentzian product ⟨·,·⟩−1; see Figure 1.1.1.

tv∞

Future light-cone

Past light-cone

Figure 1.1.1: The Lorentz-Minkowski space Rn,1

Chapter 1. Coxeter groups and Coxeter polyhedra

A vector v∈Rn,1\ {0}is said to be

•spacelike if ⟨v, v⟩−1>0,

•timelike if ⟨v, v⟩−1<0,

•lightlike if ⟨v, v⟩−1= 0.

In Figure 1.1.1, we depict a spacelike vector v, a timelike vector t, and a

lightlike vector v∞in the Lorentz-Minkowski space Rn,1.

We usually identify the hyperbolic space Hnas the upper shell Hnof the

two-sheeted hyperboloid in Rn+1, that is,

Hn={x∈Rn+1 | ⟨x, x⟩−1=−1, xn+1 >0},(1.1)

with the metric d(x, y) = dH(x, y) = arccosh(−⟨x, y⟩−1) for any x, y ∈Hn.

The boundary of Hn, denoted by ∂Hn, can then be identiﬁed with

∂Hn={x∈Rn+1 | ⟨x, x⟩1= 1,⟨x, x⟩−1= 0, xn+1 ≥0}.(1.2)

We denote the compactiﬁcation of Hnby Hn=Hn∪∂Hn.

Let us mention that in this linear model, the volume element of Hnis given

dvoln=dx1···dxn

p1 + x2

1+. . . +x2

A point p∈Hnis called an ordinary point, and a point q∈∂Hnis called

an ideal point. A metric sphere centered at an ordinary point pcarries a

spherical structure in a natural way. A metric sphere internally tangent

to an ideal point qcarries a Euclidean structure and is called a horosphere

Sqcentered at q. Horospheres as Euclidean spaces are best treated in the

upper half space model Un={x∈Rn|xn= 0}endowed with the metric

ds2

U=dx2

1+...+dx2

n. Indeed, a horosphere centered at q=∞is a hyperplane

Ht={xn=t},t > 0, with induced metric ds2

U|Ht=1

t2(dx2

1+. . . +dx2

n−1).

Let us emphasize that the group O(n, 1) of Lorentzian matrices acting by

isometries on Rn,1admits four connected components. Among them, the sub-

group O+(n, 1) of positive Lorentz-matrices preserves each of the two sheets

of the hyperboloid. It can be veriﬁed that the group IsomHnis isomorphic

to O+(n, 1).

In order to complete the picture, let us mention the well-known projective

model that will also be useful to describe Hn. For

C:= {x∈Rn+1 | ⟨x, x⟩−1<0, xn+1 >0},(1.3)

2. Hyperplanes, polyhedra and Gram matrix

we have the isometry

Hn∼

=C/R+.(1.4)

In this way, we interpret Hnas space of classes represented by timelike rays

of the future light-cone Cof Rn,1. Similarly, any point on the boundary ∂Hn

can identiﬁed with a lightlike ray belonging to ∂C.

2 Hyperplanes, polyhedra and Gram matrix

Assume that Xn=En,Snor Hn. It is well known that any element of IsomXn

can be written as a ﬁnite product of hyperplane reﬂections. Any hyperplane

in Xnseparates the space into two closed convex half-spaces, and there is a

reﬂection which ﬁxes the hyperplane and exchanges the two half spaces. In

this section, we start with the description of hyperplanes in Xn, and then

characterize convex polyhedra in Xn.

In En, a hyperplane is an aﬃne hyperplane given by a normal vector

v∈Sn−1and a translational vector u∈Enas follows.

Hu,v ={x∈En| ⟨x, v⟩1= 0}+u .

Observe that each Hu,v is isomorphic to Hv:= H0,v.

For a hyperplane Hvin En, the reﬂection svwith respect to Hvis given by

sv(x) = x−2⟨v, x⟩

||v||2v , x ∈En.(1.5)

For any two unit vectors v1, v2∈En, we say that the hyperplanes Hv1and

Hv2are parallel if and only if ⟨v1, v2⟩1=−1. For Hv1∩Hv2=∅, the dihedral

angle ∡(Hv1, Hv2) is given by

cos ∡(Hv1, Hv2) = −⟨v1, v2⟩1.

In the case of constant curvature K= 0, given a vector v∈Rn+1 such

that ⟨v, v⟩K= 1, its orthogonal complement is given by

Xv=v⊥:= {x∈Rn+1 | ⟨x, v⟩K= 0}.(1.6)

The intersection Hv:= Xv∩Xngives a hyperplane in Xnwith normal vector

v. Any hyperplane of Xnis obtained in such a way. If not otherwise stated,

we always assume vto be a unit vector. For a hyperplane Hvin Xn, we can

associate the (oriented) closed half-space

H−

v:= {x∈Xn| ⟨x, v⟩K≤0},

Chapter 1. Coxeter groups and Coxeter polyhedra

bounded by Hv, and where vis pointing outwards.

The reﬂection svwith respect to Hvgiven by

sv(x) = x−2⟨v, x⟩v , x ∈Xn,

is an isometry of Xn.

⋄If Xn=Sn, any two spherical hyperplanes Hv1, Hv2intersect in Sn. Their

dihedral angle is given by

cos ∡(Hv1, Hv2) = −⟨v1, v2⟩1.

⋄If Xn=Hn, two hyperbolic hyperplanes Hv1, Hv2intersect inside Hnif

and only if

|⟨v1, v2⟩−1|<1.

Furthermore, their dihedral angle is given by

cos ∡(Hv1, Hv2) = −⟨v1, v2⟩−1.(1.7)

Two hyperbolic hyperplanes Hv1, Hv2intersect at a point on the boundary

∂Hn, and are called (hyperbolic-)parallel, if and only if |⟨v1, v2⟩−1|= 1. This

condition is equivalent to the fact that Hv1and Hv2intersect at a dihedral

angle 0.

Lastly, two hyperplanes Hv1and Hv2do not intersect in Hn=∂Hn∪Hnif

and only if |⟨v1, v2⟩−1|>1. In this case, they are called ultraparallel, and

they give rise to a unique hyperbolic line Lorthogonal to both of them. The

distance between Hv1and Hv2is then given by

cosh d(Hv1, Hv2) = |⟨v1, v2⟩−1|.

Observe that we have ⟨v1, v2⟩−1<0 if and only if v1and v2are of opposite

orientation.

Let kbe a positive integer. For a k-dimensional vector subspace V⊂Rn,1,V

is called hyperbolic if it has nonempty intersection with Hn. More speciﬁcally,

the intersection V∩Hnis a hyperbolic (k−1)-plane. In a similar way,

V⊂Rn,1is elliptic if V∩Hnis empty, and it is parabolic in the remaining

case.

It is not diﬃcult to see but important to note that the orthogonal complement

of Vis elliptic if and only if Vis hyperbolic. In particular, a hyperplane Hv

is hyperbolic if and only if the normal vector vis spacelike.

The following elementary lemma will be useful later on.

2. Hyperplanes, polyhedra and Gram matrix

Lemma 1.2.1. Any two Lorentz-orthogonal lightlike vectors are collinear.

Proof. Assume that v, v′∈Rn,1are two Lorentz-orthogonal lightlike vectors.

Let wbe a timelike vector of squared norm −1. Then, the Lorentz-Minkowski

space Rn,1decomposes as Rw⊕w⊥. Hence, there exist λ, λ′∈R\ {0}and

two spacelike vectors x, x′∈Rn,1such that v=λw +xand v′=λ′w+x′.

Since ⟨v, w⟩=λ⟨w, w⟩+⟨x, w⟩=−λ, we derive that

⟨x, x⟩=⟨v−λw, v −λw⟩

=⟨v, v⟩ − λ2⟨w, w⟩

=λ2

and, similarly, ⟨x′, x′⟩=λ′2and ⟨x, x′⟩=λλ′. Now, one has

⟨λ′x−λx′, λ′x−λx′⟩=λ′2⟨x, x⟩+λ2⟨x′, x′⟩ − 2λλ′⟨x, x′⟩= 0 .

It follows that λ′x−λx′= 0, as λ′x−λx′belongs to the elliptic subspace

w⊥. Therefore,

λ′v−λv′=λx′−λ′x= 0 .

Now, we introduce the important concept of (convex) polyhedra in Xnas

follows.

Deﬁnition 1.2.2. Apolyhedron of dimension nor an n-polyhedron P⊂Xn

is the non-empty intersection of ﬁnitely many closed half spaces H−

viin Xn

bounded by N≥n+ 1 hyperplanes Hviin Xn, that is,

i=1

H−

vi.(1.8)

In the sequel, we often suppose P⊂Xnto be of dimension n. By con-

struction, Pis convex and entirely determined by its normal vectors up to

isometry.

The dihedral angles of Pare the angles associated with the normal vectors

of two intersecting hyperplanes in the boundary of P; see (1.7).

Depending on whether Xn=En,Snor Hn, a polyhedron P⊂Xngiven

by (1.8) is said to be Euclidean,spherical, or hyperbolic. In what follows,

we assume polyhedra to have ﬁnite volume, except if mentioned otherwise.

Observe that a polyhedron P⊂Hnis of ﬁnite volume if and only if Pis the

(hyperbolically) convex hull of ﬁnitely many points in Hn.

Chapter 1. Coxeter groups and Coxeter polyhedra

Deﬁnition 1.2.3. For 0 ≤k≤n−1, a k-face of Pis the non-empty

intersection of n−kbounding hyperplanes of Pwith Hn. A facet of Pis a

(n−1)-face of P, an edge is a 1-face of P, and an ordinary vertex is a 0-face

of P.

Deﬁnition 1.2.4. An ideal point q∈∂Hnis an ideal vertex of P⊂Hnif

q∈Pand the intersection P∩Sqof Pwith a suﬃciently small horosphere Sq

centered at qis compact when considered as an (n−1)-dimensional Euclidean

polyhedron.

Remark 1.2.5. As a consequence of the famous volume diﬀerential for-

mula of Schl¨aﬂi, the volume of a hyperbolic polyhedron Pis a monotonously

decreasing function with respect to each dihedral angle of P; see [47] for

example.

For completeness, let us introduce the f-vector of a polyhedron P⊂Xn.

Deﬁnition 1.2.6. Let P⊂Xnbe a polyhedron, and let fkbe the number

of k-faces of P. The f-vector of Pis the vector given by

(f0, . . . , fn−1).

By the celebrated formula of Euler-Sch¨aﬂi, we have the following result.

Proposition 1.2.7. Let P⊂Xnbe a polyhedron. Then,

n−1

k=0

(−1)kfk= 1 −(−1)n.

Next, we introduce a very useful tool for the characterization of a polyhedron

in Xn, its Gram matrix.

Deﬁnition 1.2.8. Let P⊂Xnbe a polyhedron as given by (1.8). The Gram

matrix Gr(P) of Pis the Gram matrix of the system of unit normal vectors

{v1, . . . , vN}given by

Gr(P)=(gij)1≤i,j≤Nwhere gij := ⟨vi, vj⟩K.

Remark 1.2.9. As deﬁned above, Gr(P) is a real symmetric matrix with

diagonal entries equal to 1. At times, we use a diﬀerent normalization of the

vectors viin order to get a matrix with integer coeﬃcients.

2. Hyperplanes, polyhedra and Gram matrix

A polyhedron Pis said to be non-degenerate if its bounding hyperplanes do

not share a common point, and there is no hyperplane orthogonal to all of

them. A non-degenerate polyhedron is said to be indecomposable if and only

if its Gram matrix Gr(P) is indecomposable in the classical sense. Observe

that any hyperbolic polyhedron of ﬁnite volume is indecomposable.

From now on we focus on acute-angled polyhedra, that is, polyhedra

whose bounding hyperplanes either intersect at a dihedral angle not exceeding

2or are disjoint.

Assume that P⊂Xnis a non-degenerate indecomposable acute-angled poly-

hedron. Depending on Xn, we have the following characterization of the

Gram matrix Gr(P).

•For Xn=En,Gr(P) is positive semi-deﬁnite of rank n.

•For Xn=Sn,Gr(P) is positive deﬁnite of rank n+ 1.

•For Xn=Hn,Gr(P) is indeﬁnite of signature (n, 1).

Observe that in the hyperbolic case, only, the matrix Gr(P) of P⊂Hncan

be arbitrarily large in comparison with n.

Assume that P⊂Xnis decomposable. For Xn=En,Gr(P) is made of

several block matrices, each one positive semi-deﬁnite of rank ki, say, but

such that Piki=n. For Xn=Sn, the Gram matrix Gr(P) is made of

several positive deﬁnite block matrices of rank kisuch that Piki=n+ 1.

If a polyhedron P⊂Xnis bounded by precisely N=n+ 1 hyperplanes, P

is called a simplex.

As a consequence of the Perron-Frobenius theory, we have the following re-

sult.

Theorem 1.2.10. Let nbe a positive integer. Let G= (gij )1≤i,j≤n+1 ∈

Mat(n+ 1,R)be an indecomposable symmetric matrix such that gii = 1 for

i= 1, . . . , n + 1 and such that −1≤gij ≤0for 1≤i=j≤n+ 1. Then,

one has the following properties.

•If Gis positive semi-deﬁnite of rank nsuch that, for all 1≤i, j ≤n+1,

the (i, j)-th cofactor of Gis positive, then Gis the Gram matrix of a

Euclidean simplex in Enwhich is unique up to isometry.

•If gij =−1for all 1≤i, j ≤n+ 1,i=j, and Gis positive deﬁnite,

then Gis the Gram matrix of a spherical simplex in Snwhich is unique

up to isometry.

Chapter 1. Coxeter groups and Coxeter polyhedra

•If Ghas signature (n, 1) such that, for all 1≤i, j ≤n+ 1, the (i, j)-

th cofactor of Gis positive, then Gis the Gram matrix of a compact

hyperbolic simplex in Hnwhich is unique up to isometry.

For the cases Xn=Snand En, we cite the following additional results.

Theorem 1.2.11. Any acute-angled non-degenerate spherical (respectively,

Euclidean) polyhedron Pis a simplex (respectively, a direct product of sim-

plices).

Let P⊂Hnbe a polyhedron and k≥1. Each k-face Fof Pgives rise to a

principal submatrix Gr(F) of the matrix Gr(P). Here, Gr(F) is the matrix

formed by the columns and rows corresponding to the facets of Pcontaining

F. As a consequence, there is a one-to-one correspondence between the set

of k-faces of Pand the set of positive deﬁnite principal submatrices of Gr(P)

of rank k. A similar result holds for ordinary vertices as they correspond to

positive deﬁnite principal submatrices of rank nof Gr(P).

Let P⊂Hnbe acute-angled and of ﬁnite volume. An ideal vertex of Pgives

rise to a neighbourhood in Pthat is a cone over a direct product of simplices.

As a consequence, there is a one-to-one correspondence between the set of

ideal vertices of Pand the set of positive semi-deﬁnite principal submatrices

of rank n−1 of Gr(P).

Finally, a vertex of a polyhedron P⊂Xnis said to be simple if it is the

intersection of precisely nbounding hyperplanes of P. The polyhedron Pis

called simple if all of its vertices are simple. It follows from the above that

any compact acute-angled hyperbolic polyhedron is simple.

3 Abstract and geometric Coxeter groups

Deﬁnition 1.3.1. An abstract Coxeter group Γ = (W, S) of rank Nis a

ﬁnitely generated group Wwith generating set Sof cardinality Nand a

presentation according to

W=⟨si∈S|s2

i= 1,(sisj)mij = 1⟩,(1.9)

where mij =mji ∈ {2,3,··· ,∞} for all i=j.

Observe that the set Sof generators of a Coxeter group is symmetric, that

is, S=S−1.

3. Abstract and geometric Coxeter groups

Let Γ = (W, S) be an abstract Coxeter group. For any subset T⊂S, the

group WTgenerated by the elements of Tis a subgroup of W, and WTis

itself a Coxeter group. We say that WTis a parabolic subgroup of W.

We are interested in Coxeter groups that admit a geometric representation

as discrete subgroups of IsomXngenerated by hyperplane reﬂections.

A natural approach is to consider a polyhedral arrangement of hyperplanes

as follows.

Deﬁnition 1.3.2. ACoxeter polyhedron P=TN

i=1 H−

vi⊂Xnis a polyhedron

all of whose dihedral angles are zero or of the form π

kfor k∈Z≥2.

Given a Coxeter polyhedron P⊂Xnas in Deﬁnition 1.3.2, we denote

by s1, . . . , sN∈IsomXnthe reﬂections with respect to the hyperplanes

Hv1, . . . , HvNbounding P. Let Γ be the group generated by S={s1, . . . , sN}.

If two hyperplanes Hvi, Hvjintersect at a dihedral angle π

mij , then the compo-

sition sisjis a rotation of angle 2π

mij . Furthermore, if two hyperplanes do not

intersect in Hn(that is, they are parallel or ultraparallel), the composition

sisjhas inﬁnite order. In this way, we deduce that Γ admits a presentation

as a Coxeter group

Γ = ⟨s1, . . . , sN|s2

i= 1,(sisj)mij = 1⟩.(1.10)

Deﬁnition 1.3.3. Let P⊂Xnbe a Coxeter polyhedron. The Coxeter

group Γ ⊂IsomHngenerated by reﬂections with respect to the bounding

hyperplanes of Pis called the geometric Coxeter group associated with P.

The Gram matrix Gr(Γ) of Γ is deﬁned to be the Gram matrix Gr(P) of P.

Depending on whether Xn=En,Snor Hn, the geometric Coxeter group Γ

is said to be Euclidean,spherical, or hyperbolic. Observe that a spherical

Coxeter group is always ﬁnite.

The Coxeter group Γ ⊂IsomXnis called cocompact if the associated Coxeter

polyhedron P⊂Xnis compact, and non-cocompact otherwise. In addition,

Γ is called coﬁnite if the associated Coxeter polyhedron Phas ﬁnite volume.

In what follows, we assume hyperbolic Coxeter groups to be coﬁnite, except

if mentioned otherwise.

Chapter 1. Coxeter groups and Coxeter polyhedra

3.1 The Coxeter diagram of a Coxeter group

Now, we introduce the notion of Coxeter diagram for an abstract Coxeter

group Γ = (W, S) of rank Npresented by

W=⟨si∈S|s2

i= 1,(sisj)mij = 1⟩.

Deﬁnition 1.3.4. The Coxeter diagram Σ of Γ is the (non-oriented) graph

whose nodes correspond to the generators s1, . . . , sNof Γ. Any two nodes

νiand νjin the graph are connected by an edge labelled by mij when the

corresponding generators satisfy (sisj)mij = 1 for mij ∈ {3,...,∞}. In other

words, two nodes are not connected if the corresponding generators commute.

We omit the label when mij = 3 since it occurs frequently.

If a Coxeter diagram is a linear graph with consecutive edges labelled by

k1, . . . , kr, we describe the corresponding Coxeter group (or Coxeter polyhe-

dron) by its Coxeter symbol [k1, . . . , kr].

Each parabolic subgroup of (W, S) gives rise to a subdiagram σof Σ. Note

that Coxeter diagrams and/or some of their subdiagrams need not to be

connected graphs. However, a Coxeter diagram Σ is connected if and only if

its Coxeter group (W, S) is irreducible.

In the case of a geometric Coxeter group Γ ⊂IsomXn, its Coxeter diagram

Σ is called aﬃne,spherical, or hyperbolic, depending on whether Xn=En,Sn

or Hn.

Furthermore, in this geometric context and by (1.8) and (1.9), the Coxeter

diagram Σ coincides with the graph whose nodes correspond to the bounding

hyperplanes of the associated Coxeter polyhedron P⊂Xn, where two nodes

are joined by an edge labelled by mij when the corresponding hyperplanes

share a dihedral angle π

mij for mij ≥3. An edge is labelled by ∞when the

corresponding hyperplanes are disjoint inside Xnfor Xn=Sn.

Example 1.3.5. A simple but prominent example is given by the Coxeter

group Γ with Coxeter symbol [7,3] based on the hyperbolic Coxeter triangle

∆(π

2,π

3,π

7). In fact, the group Γ is of minimal co-area among all discrete

groups in IsomH2, and has minimal growth rate given by Lehmer’s number;

see Chapters 3 and 5.

In the sequel, for a geometric Coxeter group Γ, we do not distinguish between

Γ, its Coxeter polyhedron P, the Coxeter diagram Σ, and the Gram matrix

Gr(Γ).

3. Abstract and geometric Coxeter groups

The number Nof nodes is called the order of the Coxeter diagram Σ, and it

coincides with the rank of Γ. The rank of the Gram matrix Gr(Γ) is called

the rank of Σ.

Note that if the polyhedron Pis indecomposable, its Coxeter diagram Σ is

connected and the group Γ is irreducible, and vice versa.

3.2 Spherical and Euclidean Coxeter groups

In the fundamental work [16], H. S. M. Coxeter classiﬁed all spherical and

Euclidean Coxeter groups. In Tables 1.3.1 and 1.3.2, we give the list of all

irreducible spherical and Euclidean Coxeter groups in terms of their Coxeter

diagrams.

n≥1Ann= 3 H3

n≥2Bn

4n= 4 H4

n≥3Dnn= 6 E6

n= 2 G(m)

mn= 7 E7

n= 4 F4

4n= 8 E8

Table 1.3.1: The irreducible spherical Coxeter groups of rank n

The following theorems have been established by Coxeter [16]; see also the

work of Vinberg [84].

Theorem 1.3.6. Let P⊂Sn−1be a spherical Coxeter polyhedron with asso-

ciated Coxeter group Γ. Then, Pis a simplex, and its Coxeter diagram is a

disjoint union of Coxeter diagrams of irreducible spherical Coxeter groups of

rank kisuch that Pki=n.

In contrast to the spherical case, by Theorem 1.2.10, any connected aﬃne

Coxeter diagram of rank nhas order n+ 1.

Theorem 1.3.7. Let P⊂Enbe a Euclidean Coxeter polyhedron of ﬁnite

volume with associated Coxeter group Γ. Then, Pis a product of simplices,

and its Coxeter diagram is a disjoint union of Coxeter diagrams of irreducible

aﬃne Coxeter groups of rank kisuch that Pki=n.

Chapter 1. Coxeter groups and Coxeter polyhedra

n= 1 e

A1∞n= 2 e

n≥2e

Ann= 4 e

n≥3e

n= 6 e

n≥2e

4 4 n= 7 e

n≥4e

Dnn= 8 e

Table 1.3.2: The irreducible aﬃne Coxeter groups of rank n

3.3 Hyperbolic Coxeter groups

Let P=TN

i=1 H−

vibe a Coxeter polyhedron in Hnand Γ its associated Coxeter

group of rank N. Assume that all the vectors vinormal to the bounding

hyperplanes Hi=Hviof Phave Lorentzian norm 1.

The Gram matrix Gr(Γ) ∈Mat(N, R) of Γ (and of the polyhedron P) has

the following coeﬃcients oﬀ the diagonal; see Section 2.

⟨vi, vj⟩−1=





−cos π

mij if ∡(Hi, Hj) = π

mij = 0

−1 if Hi, Hjare parallel

−cosh lif dH(Hi, Hj) = l > 0

(1.11)

Vertices and faces

In what follows we describe vertices and faces of a hyperbolic Coxeter polyhe-

dron and cite important correspondences at the level of its Coxeter diagram.

Theorem 1.3.8. Let Pbe a Coxeter polyhedron in Hn, and let Σbe its

Coxeter diagram. Then, for 1≤k≤n, there is a one-to-one correspondence

between (n−k)-faces of Pand spherical subdiagrams σof Σof rank k.

In particular, any ordinary vertex corresponds to a spherical subdiagram of

rank n. In addition, ideal vertices of Pcorrespond to aﬃne subdiagrams σ∞

of rank n−1.

Let us add that an ideal vertex of a hyperbolic Coxeter polyhedron is non-

simple if and only if σ∞is made of at least two aﬃne components.

3. Abstract and geometric Coxeter groups

By Theorem 1.3.8, a spherical subdiagram σof rank kof Σ corresponds to

an (n−k)-face F=F(σ) of P; see Table 1.3.1. The face Fitself is an

acute-angled polyhedron of ﬁnite volume, but it is not necessarily a Coxeter

polyhedron in Hn−k.

The following important theorem provides a condition for the face Fto

be a Coxeter polyhedron. It was ﬁrst proven by Borcherds [5] in a more

general context, and has been reformulated by Allcock [1] as follows.

Theorem 1.3.9. Let Pbe a hyperbolic Coxeter polyhedron with Coxeter dia-

gram Σ. Let F=F(σ)be a face of Pcorresponding to a spherical subdiagram

σof Σ. Assume that σdoes not contain any component of type Al, for l≥1,

or D5. Then, Fis itself a Coxeter polyhedron.

The facets of F⊂Pcorrespond to all those nodes of Σ that, together with

σ, form a spherical subdiagram of Σ. We say that a node ν∈Σ\σis a good

neighbour of σif the subdiagram spanned by σand νis spherical. Otherwise,

νis said to be a bad neighbour. As a consequence, the facets of Fcorrespond

to the good neighbours of σ.

Let ν1, ν2be two good neighbours of σ. Then, the nodes ν1, ν2correspond

to facets f1=F1∩F, f2=F2∩Fof F, where F1, F2are facets of P. Their

dihedral angle ∡(f1, f2) is less than or equal to ∡(F1, F2).

More precisely, according to Allcock [1], the dihedral angles ∡(f1, f2) of the

facets f1, f2of Fare given as follows.

Theorem 1.3.10. Let Pbe a Coxeter polyhedron, and let F=F(σ)be

a face of Pwith facets f1, f2as above. Assume that σdoes not contain

any component of type Al, for l≥1, or D5. Then, one has the following

characterization.

1. If neither ν1nor ν2attaches to σ, then ∡(f1, f2) = ∡(F1, F2).

2. Assume ν1, ν2attach to diﬀerent components of σ. If ∡(F1, F2) = π

then ∡(f1, f2) = π

2. Otherwise, f1and f2are disjoint.

3. Assume ν1, ν2attach to the same component σ0of σ. If ν1and ν2are

not joined by an edge and {σ0, ν1, ν2}yields a diagram E6(respectively,

E8or F4), then ∡(f1, f2) = π

3(respectively, π

4). Otherwise, f1and f2

are disjoint.

4. Assume ν1attaches to a component σ0of σ, and ν2does not attach to

σ. If ∡(F1, F2) = π

2, then ∡(f1, f2) = π

2. If ν1and ν2are joined by a

simple edge and {σ0, νi, νj}yields a diagram Bk(respectively, Dk,E8

Chapter 1. Coxeter groups and Coxeter polyhedra

or H4), then ∡(f1, f2) = π

4(respectively, π

4,π

6or π

10 ). Otherwise, f1

and f2are disjoint.

Remark 1.3.11. Let m≥6 be an integer. Then, any node connected to

the subdiagram G(m)

2= [m] of Σ is a bad neighbour. As a result of Theorem

1.3.9, it is easy to identify each (n−2)-face F(G(m)

2) of a hyperbolic Coxeter

polyhedron P⊂Hn. Furthermore, by Theorem 1.3.10, its dihedral angles

are precisely given by the dihedral angles of P.

Example 1.3.12. Let P⊂H5be the Coxeter pyramid depicted below.

6 6

By Theorem 1.3.9 and Remark 1.3.11, each 3-face of Pcorresponding to

one of the two subdiagrams G2=G(6)

2= [6] is a Coxeter tetrahedron with

Coxeter symbol [6,3,3] (or [3,3,6]) .

Example 1.3.13. For the Coxeter 7-pyramid depicted below, the 5-face

F(G2) is a pyramid over a product of two simplices of type e

A2; see also

Table 3.2.2.

Criteria for compactness and ﬁnite volume

Recall that a hyperbolic Coxeter n-polyhedron Phas ﬁnite volume if it is the

convex hull of a ﬁnite number of points in Hn. The polyhedron Pis compact

if all of its vertices are ordinary points in Hn. Otherwise, Pis non-compact

(but of ﬁnite volume) and has at least one ideal vertex; see Deﬁnition 1.2.4.

The following criteria were established by Vinberg [82].

Theorem 1.3.14 (Compactness criterion).A polyhedron P⊂Hnis compact

if and only if the following holds.

1. Phas at least one ordinary vertex.

2. For every vertex vof Pand every edge of Pemanating from v, there

is precisely one other vertex of Pon that edge.

3. Abstract and geometric Coxeter groups

Theorem 1.3.15 (Finite-volume criterion).A polyhedron P⊂Hnhas ﬁnite

volume if and only if the following holds.

1. Phas at least one (ordinary or ideal) vertex.

2. For every (ordinary or ideal) vertex vof Pand every edge of Pema-

nating from v, there is precisely one other (ordinary or ideal) vertex of

Pon that edge.

For a Coxeter polyhedron Pwith Coxeter diagram Σ, Theorem 1.3.15 can

be reformulated as follows.

1. Σ contains at least one spherical subdiagram of rank nor one aﬃne

subdiagram of rank n−1.

2. Each spherical subdiagram of rank n−1 of Σ can be extended in exactly

two ways to a spherical subdiagram of rank nor to an aﬃne subdiagram

of rank n−1.

Remark 1.3.16. By exploiting Theorems 1.3.14 and 1.3.15, Guglielmetti

[30, 31] developed the software CoxIter, which, for a given hyperbolic Cox-

eter diagram Σ = Σ(P), indicates whether the polyhedron Pis compact or

of ﬁnite volume. The software CoxIter provides some more important infor-

mation. For example, the Euler characteristic of the group Γ(P) is displayed,

which, in the even dimensional case, is well known to be proportional to the

volume of P.

Finally, let us add another non-trivial result, due to Felikson and Tu-

markin [25], which will be very useful for us in the sequel; see Chapter 4.

Proposition 1.3.17. Let Pbe a hyperbolic Coxeter n-polyhedron (of ﬁnite

volume), and let Σbe its Coxeter diagram. Then, no proper subdiagram of

Σis the Coxeter diagram of a ﬁnite-volume Coxeter n-polyhedron.

Arithmeticity of hyperbolic Coxeter groups

The general theory of arithmetic groups is a broad ﬁeld which we present

here only brieﬂy and this in the restricted case of hyperbolic Coxeter groups.

We refer to [57, 58, 81], for example.

We begin by the following deﬁnition, comparing two discrete groups Γ1,Γ2

in IsomHn. The groups Γ1and Γ2are commensurable (in the wide sense) if

there is an element γ∈IsomHnsuch that Γ1∩γΓ2γ−1has ﬁnite index in both

Chapter 1. Coxeter groups and Coxeter polyhedra

Γ1and γΓ2γ−1. Commensurability preserves properties such as discreteness

and ﬁnite covolume.

A fundamental theorem of Margulis states that a coﬁnite group Γ in IsomHn,

n≥3 is non-arithmetic if and only if its commensurator

Comm(Γ) = {γ∈Isom Hn|Γ and γΓγ−1are commensurable }

is a discrete subgroup of ﬁnite covolume in IsomHn; see [87], for example.

Margulis’ result can be taken as a deﬁnition of a coﬁnite discrete group

Γ⊂IsomHnto be (non-)arithmetic.

There are several arithmeticity criteria and results. For instance, in dimen-

sion n= 2, Takeuchi [58, Appendix 13.3] classiﬁed all discrete triangle groups

up to arithmeticity.

Example 1.3.18. The four cocompact triangle groups (p, q, r) where p, q, r ∈

{2,3,6}and with at least one entry equal to 6, are all arithmetic.

Let Γ ⊂IsomHnbe a non-cocompact Coxeter group of ﬁnite covolume. In

this restricted context, the arithmeticity property can be characterized in an

easy way. For its formulation, we consider the Gram matrix Gr(Γ) of Γ and

its related cycles.

Deﬁnition 1.3.19. For an arbitrary matrix A= (aij )i,j∈{1,...,N}, a cycle in

Ais an element of the form

ai1i2···aik−1ikaiki1for i1, . . . , ik∈ {1, . . . , N}, k ≥2.

A cycle in Ais said to be irreducible if all i1, . . . , ikare distinct.

The following arithmeticity criterion is due to Vinberg [84].

Theorem 1.3.20 (Arithmeticity criterion).Let Γ⊂IsomHnbe a non-

cocompact Coxeter group of ﬁnite covolume. Denote its Gram matrix by

G=Gr(Γ). Then, Γis arithmetic (and deﬁned over Q) if and only if all

cycles in 2Gare rational integers.

In view of Chapter 4, we specialize the context even more, and consider hy-

perbolic Coxeter groups Γ as in (1.9) which satisfy the conditions of Theorem

1.3.20, and whose Coxeter polyhedra P⊂Hnhave no pair of ultraparallel

facets. Then, Guglielmetti [32] added the following characterization.

Theorem 1.3.21. Let P⊂Hnbe a non-compact Coxeter polyhedron with

no pair of ultraparallel facets. Then, its Coxeter group Γis arithmetic if and

only if mij ∈ {2,3,4,6,∞} for all distinct i, j, and any irreducible cycle in

2Gof length at least 3is an integer.

3. Abstract and geometric Coxeter groups

The following example gives an illustration of Theorem 1.3.21.

Example 1.3.22. The Coxeter pyramid group [6,3,3,3,3,6] in IsomH5is

arithmetic. However, the Coxeter 5-simplex group depicted in Figure 1.3.22

is non-arithmetic.

Figure 1.3.2: The non-arithmetic Coxeter simplex group in IsomH5

CHAPTER 2

Root systems and fundamental weights

In this chapter, we provide a brief survey of the classical theory of root

systems in Euclidean vector spaces. Their classiﬁcation including details

about their simple root systems, highest root and fundamental weights will

play an important role in Chapter 4.

As references for this chapter, we quote [6, 37].

1 Root systems and Weyl groups

From now on, we denote by Va Euclidean vector space of ﬁnite dimension.

Deﬁnition 2.1.1. Aroot system Ris a ﬁnite subset of Vmade of non-zero

vectors, called roots, such that the following conditions hold.

1. Rgenerates V.

2. R∩Rα={α, −α}.

3. For any root α∈R, the reﬂection sαalong α∈Ras given by (1.5)

permutes the elements of R, that is, sα(β)∈Rfor all β∈R.

The rank of the root system Ris deﬁned to be the dimension of V. The root

system Ris reducible if there exist non-empty, disjoint root systems R1, R2

such that R=R1∪R2and span(R) = span(R1)⊕span(R2). Otherwise, the

root system Ris said to be irreducible.

1. Root systems and Weyl groups

We often impose the crystallographic condition for R, ensuring that the vector

sα(β) is obtained from βby adding an integral multiple of α.

Deﬁnition 2.1.2 (Crystallographic condition).Aroot system Ris said to

be crystallographic if for any roots α, β ∈Rone has

kα,β := 2⟨α, β⟩

⟨α, α⟩∈Z.(2.1)

In fact, it is suﬃcient to require the condition (2.1) for a set of simple roots

in Ras follows. Let us ﬁrst introduce the notion of positive roots.

Let R+⊂Rsatisfy the following properties.

1. For each α∈R, exactly one of αand −αbelongs to R+;

2. For any distinct α, β ∈R+such that α+β∈R, one has α+β∈R+.

The elements of R+are called positive roots.

Deﬁnition 2.1.3. Asimple root is a positive root that cannot be decom-

posed as the sum of two positive roots. We denote by B=B(R) the set of

simple roots.

Note that every system R+of positive roots contains a unique set Bof simple

roots. Furthermore, the set Bof simple roots is a basis of V.

Deﬁnition 2.1.4. The group W=W(R) generated by the reﬂections sα

through the hyperplanes α⊥associated with the roots α∈Bis called the

Weyl group of R.

Proposition 2.1.5. The Weyl group Wis ﬁnite.

For a ﬁxed system Bof simple roots in R, the Weyl group Wsatisﬁes the

relations

(sαsβ)mα,β = 1 for α, β ∈B, (2.2)

where mα,β is the order of the composition sαsβin W. Hence, Wcan be

identiﬁed with a spherical Coxeter group; see Section 3.2 in Chapter 1.

Let us mention the following two well-known results.

Proposition 2.1.6. Any positive root β∈R+can be written as a linear

combination of simple roots, that is,

β=X

α∈B

mαα , (2.3)

where each mαis a non-negative integer.

Chapter 2. Root systems and fundamental weights

Proposition 2.1.7. For any two distinct simple roots α, β ∈B, one has

⟨α, β⟩ ≤ 0.(2.4)

We consider the following natural partial ordering on Rwith respect to R+.

For two roots α, β ∈R, one says that α≤βif and only if β−αis a sum of

simple roots with non-negative coeﬃcients; see Proposition 2.1.6.

If Ris an irreducible root system, there exists a unique highest root with

respect to this ordering, and we will denote it by ¯α. In view of Proposition

2.1.7, the highest root ¯αforms a non-obtuse angle with every simple root

β∈B.

From now on, we consider a crystallographic root system R; see (4).

As ⟨α, β⟩=||α|| · ||β||cos∡(α, β), it follows that

kα,βkβ,α = 4cos2∡(α, β)∈Z≥0.(2.5)

Since kα,β and kβ,α are integers, one gets

kα,βkβ,α ∈ {0,1,2,3,4}.(2.6)

The crystallographic condition for Rimplies that its simple roots can be of

at most two diﬀerent lengths, and they are called short and long roots. In

fact, the quotient of the squared lengths of two simple roots can be equal to

2 or 3, only.

Furthermore, the Weyl group Wgenerated by sα,α∈B, is a ﬁnite reﬂection

group with mα,β ∈ {2,3,4,6}for any two distinct simple roots α, β ∈B;

see (2.2). The group Wis said to be crystallographic, and Wstabilizes the

Z-lattice spanned by B.

Deﬁnition 2.1.8. Let Λ be the set of vectors w∈Vsuch that kw,α ∈Z

for all α∈R. The elements of Λ are called weights, and a weight w∈Λ is

dominant if all the integers kw,α are nonnegative for all α∈R+.

Lastly, we deﬁne the notion of fundamental weights as follows. Let nbe the

rank of R, and write B={α1, . . . , αn}.

Deﬁnition 2.1.9. The fundamental weights w1, . . . , wn∈Λ are the domi-

nant weights deﬁned by the condition kwi,αj=δij for all 1 ≤i, j ≤n, where

δij is the Kronecker symbol.

2. Dynkin diagrams and extended Dynkin diagrams

2 Dynkin diagrams and extended Dynkin diagrams

Let Rbe an irreducible crystallographic root system in V, and let W=W(R)

be its Weyl group.

The Dynkin diagram of such a root system encodes all the information about

the relative root lengths of simple roots of Rand their angles. In the fol-

lowing, we restrict the context to the subfamily of Weyl groups used in this

work, that is, the Weyl groups of type A, D, E and G2.

In this context, the Dynkin diagram of Rwill coincide with the Coxeter

diagram of Wexcept for the case G2.

Deﬁnition 2.2.1. The Dynkin diagram of Ris the graph whose nodes corre-

spond to the simple roots of R, and where two nodes are joined as follows. If

the roots are orthogonal, there is no edge between the corresponding nodes.

If the angle between the two roots is 2π

3, there is a simple edge between the

nodes. If the angle is 5π

6, the two nodes are joined by a triple edge, and there

is an arrow on the edge pointing from the long root to the short root.

When all roots have the same length, the Dynkin diagram has simple edges

and is called simply laced. It corresponds to one of the Coxeter diagrams

occurring in Figure 1.3.1.

Note that a root system is irreducible if and only if its Dynkin diagram is

connected.

The extended Dynkin diagram of Rcontains additional information in view

of an extra root added to the set Bof simple roots of Ras follows.

Deﬁnition 2.2.2. The extended Dynkin diagram of Ris the diagram ob-

tained by adding the node to the Dynkin diagram of Rthat corresponds to

the inverse −¯αof the highest root ¯α.

Since ¯αis a linear combination of the simple roots of R, the extended Dynkin

diagram is closely related to a connected aﬃne Coxeter diagram; see Table

1.3.2.

3 Classiﬁcation of root systems of type A,D,Eand G2

In this section, we consider some particular irreducible crystallographic root

systems. We choose a natural system of simple roots, establish the highest

root and provide the fundamental weights. In addition, we depict the associ-

ated extended Dynkin diagram decorated by the corresponding root at each

node.

Chapter 2. Root systems and fundamental weights

Let us ﬁrst introduce the ADE-root systems. Their Dynkin diagrams are

simply laced.

Denote by ϵithe i-th vector of the standard basis of Rk.

Deﬁnition 2.3.1. Let Vbe the subspace of vectors in Rn+1 whose coordi-

nates sum up to 0. The An-root system Anis the set of vectors in Vof length

√2 with integer coordinates given by

An={ϵi−ϵj|1≤i=j≤n+ 1}.

The natural simple roots are α1=ϵ1−ϵ2, . . . , αn=ϵn−ϵn+1. Then, the

highest root is given by α=α1+. . .+αn=ϵn+1 −ϵ1=: −αn+1. The notation

αn+1 =−¯αwill also be used in the subsequent cases. The fundamental

weights of Ancan be expressed as

wi=ϵ1+. . . +ϵi−i

n+ 1

n+1

j=1

ϵjfor 1 ≤i≤n .

The extended Dynkin diagram is given as follows, where each node is indexed

by the corresponding root.

αn+1

α1α2αn−1αn

Deﬁnition 2.3.2. Let V=Rn. The Dn-root system Dnis the set of vectors

in Vof length √2 with integer coordinates given by

Dn={ϵi−ϵj|1≤i=j≤n} ∪ {±(ϵi+ϵj)|1≤i<j≤n},

and the natural simple roots are α1=ϵ1−ϵ2, . . . , αn−1=ϵn−1−ϵn,αn=

ϵn−1+ϵn. The highest root satisﬁes

α=α1+ 2α2+. . . + 2αn−2+αn−1+αn=ϵ1+ϵ2,

and the fundamental weights are given by







wi=ϵ1+. . . +ϵifor i<n−1

wn−1=1

2(ϵ1+. . . +ϵn−1−ϵn)

wn=1

2(ϵ1+. . . +ϵn−1+ϵn)

3. Classiﬁcation of root systems of type A,D,Eand G2

The extended Dynkin diagram is given as follows.

α1

αn+1 α2αn−2

αn−1

αn

Deﬁnition 2.3.3. Let V=R8, and consider the lattice L=L0+Z(1

1≤i≤8

ϵi)

where L0consists of all vectors P

1≤i≤8

ciϵiwith ci∈Zand P

1≤i≤8

cieven.

The E8-root system E8is the set of vectors of length √2 in L. One has

E8={±ϵi±ϵj|i<j}∪{1

i=1

(−1)ηiϵi|

i=1

ηiis even },

and the natural simple roots are given by







α1=1

2(ϵ1−ϵ2− ··· − ϵ7+ϵ8)

α2=ϵ1+ϵ2

αi=ϵi−1−ϵi−2for 3 ≤i≤8

The highest root can be expressed as

α= 2α1+ 3α2+ 4α3+ 6α4+ 5α5+ 4α6+ 3α7+ 2α8=ϵ7+ϵ8,

and the fundamental weights are











w1= 2ϵ8

w2=1

2(ϵ1+ϵ2+ϵ3+ϵ4+ϵ5+ϵ6+ϵ7+ 5ϵ8)

w3=1

2(−ϵ1+ϵ2+ϵ3+ϵ4+ϵ5+ϵ6+ϵ7+ 7ϵ8)

w4=ϵ3+ϵ4+ϵ5+ϵ6+ϵ7+ 5ϵ8

w5=ϵ4+ϵ5+ϵ6+ϵ7+ 4ϵ8

w6=ϵ5+ϵ6+ϵ7+ 3ϵ8

w7=ϵ6+ϵ7+ 2ϵ8

w8=ϵ7+ϵ8

The extended Dynkin diagram is given as follows.

α3α4α5α6α7

α2

α8α9

α1

Chapter 2. Root systems and fundamental weights

Deﬁnition 2.3.4. Let Vbe the hyperplane in R8generated by the ﬁrst seven

simple roots α1, . . . , α7of E8; see Deﬁnition 2.3.3. Then, Vis orthogonal to

w8. The E7-root system E7is deﬁned as E7=E8∩V. One has

E7={±ϵi±ϵj|1≤i≤j≤6}∪{ ±(ϵ7−ϵ8)}∪{1

i=1

(−1)ηiϵi+ϵ7−ϵ8|

i=1

ηiis odd},

and the natural simple roots are α1, . . . , α7. The highest root is given by

α= 2α1+ 2α2+ 3α3+ 4α4+ 3α5+ 2α6+α7=ϵ8−ϵ7,

and the fundamental weights are











w1=−ϵ7+ϵ8

w2=1

2(ϵ1+ϵ2+ϵ3+ϵ4+ϵ5+ϵ6−2ϵ7+ 2ϵ8)

w3=1

2(−ϵ1+ϵ2+ϵ3+ϵ4+ϵ5+ϵ6−3ϵ7+ 3ϵ8)

w4=ϵ3+ϵ4+ϵ5+ϵ6−2ϵ7+ 2ϵ8

w5=ϵ4+ϵ5+ϵ6−3

2ϵ7+3

2ϵ8

w6=ϵ5+ϵ6−ϵ7+ϵ8

w7=ϵ6−1

2ϵ7+1

2ϵ8

The extended Dynkin diagram is given as follows.

α1α3α4α5α6

α2

α7

α8

Deﬁnition 2.3.5. Let Vbe the hyperplane generated by α1, . . . , α6in R8.

Then, Vis orthogonal to w7and w8. The E6-root system E6is deﬁned as

E6=E8∩Vand has the natural simple roots α1, . . . , α6.

The highest root is given by

α=α1+ 2α2+ 2α3+ 4α4+ 2α5+α6,

and the fundamental weights are











w1=2

3(−ϵ6−ϵ7+ϵ8)

w2=1

2(ϵ1+ϵ2+ϵ3+ϵ4+ϵ5−ϵ6−ϵ7+ϵ8)

w3=1

2(−ϵ1+ϵ2+ϵ3+ϵ4+ϵ5) + 5

6(−ϵ6−ϵ7+ϵ8)

w4=ϵ3+ϵ4+ϵ5−ϵ6−ϵ7+ϵ8

w5=ϵ4+ϵ5+2

3(−ϵ6−ϵ7+ϵ8)

w6=ϵ5+1

3(−ϵ6−ϵ7+ϵ8)

3. Classiﬁcation of root systems of type A,D,Eand G2

The extended Dynkin diagram is given as follows.

α1α3α4α5α6

α2

α7

Lastly, we introduce the G2-root system G2, whose natural simple roots will

have diﬀerent lengths.

Deﬁnition 2.3.6. Let Vbe the hyperplane in R3consisting of all vectors

whose coordinates add up to 0. The G2-root system G2is the set of vectors

in Vwith integer coordinates and of length √2 or √6 given by

G2={±(ϵi−ϵj)|1≤i<j≤3} ∪ {±(2ϵi−ϵj−ϵk)|1≤i,j,k≤3},

and the natural simple roots are α1=ϵ1−ϵ2, α2=−2ϵ1+ϵ2+ϵ3. Then, the

highest root can be written according to

α= 3α1+ 2α2=−ϵ1−ϵ2+ 2ϵ3,

and the fundamental weights are given by w1= 2α1+α2and w2= 3α1+2α2.

α3α2α1

We do not discuss the crystallographic root systems of type F4,Bnand Cn

as they will not be used in this work; for more details about them, see [37].

We adore chaos because we love to

produce order.

M.C. Escher

Part II:

On the existence of

hyperbolic Coxeter groups

CHAPTER 3

Some classification results

The classiﬁcation of hyperbolic Coxeter polyhedra is far from being complete.

In fact, for dimensions beyond 3, only a comparatively small quantity of them

is known.

We begin this chapter by quoting some general non-existence results for hy-

perbolic Coxeter polyhedra of ﬁnite volume. Then, we present the most

important classiﬁcation results for hyperbolic Coxeter polyhedra serving for

our purposes. The ﬁrst section is a brief survey of some of the results in

dimensions 2 and 3. The second section is devoted to hyperbolic Coxeter

polyhedra in higher dimensions.

Main references for this chapter are [6, 15, 21, 24, 37, 70, 84]; see also the

webpage survey of Felikson [23].

The following non-existence results for hyperbolic Coxeter polyhedra have

been established by Khovanski [49] and Prokhorov [69] for the ﬁnite-volume

case, and by Vinberg [83] in the compact case.

Theorem 3.0.1. Hyperbolic Coxeter polyhedra of ﬁnite volume do not exist

in dimensions bigger than 995.

Theorem 3.0.2. Compact hyperbolic Coxeter polyhedra do not exist in di-

mensions bigger than 29.

However, in dimensions beyond 3, examples of hyperbolic Coxeter polyhedra

are known for dimensions ≤19 and in dimension 21 in the ﬁnite volume case,

and for dimensions ≤8 in the compact case, only.

Chapter 3. Some classiﬁcation results

Note that throughout this chapter, and as usually, Coxeter polyhedra are

considered to have ﬁnite volume.

1 In low dimensions

In dimensions 2 and 3, there exist inﬁnitely many Coxeter polyhedra. There

are fundamental results, proven by Poincar´e and Andreev, that characterize

their existence.

In dimension 2

Theorem 3.1.1 (Poincar´e [68]).Let N∈Z≥3. A hyperbolic (convex) N-gon

P⊂H2with angles αiexists if and only if

1≤i≤N

αi< π(N−2).

From Theorem 3.1.1, we deduce the following for Coxeter polygons. For

N= 3, there exist inﬁnitely many Coxeter triangles in H2, as there are

inﬁnitely many integers m1, m2, m3≥2 such that

<1.

The Coxeter diagram of a compact Coxeter triangle is then given by

where 1

<1.

For a non-compact Coxeter triangle, the Coxeter diagram has the form

∞

for (arbitrarily large) m1≥2 and m2≥3.

Example 3.1.2. As a consequence of Theorem 3.1.1, there exist no compact

hyperbolic squares, as the sum of the angles should not exceed 2π. However,

there exist right-angled N-gons in H2for any N≥5.

1. In low dimensions

Remark 3.1.3. Observe that for N≥4, a Coxeter N-gon admits at least

one pair of ultraparallel edges.

Let Pbe a hyperbolic triangle with angles α1, α2and α3. Then, the

well-known defect formula for the area of Pyields

area(P) = π−(α1+α2+α3).

Let us situate hyperbolic Coxeter triangles in the context of fundamental

polygons of discrete groups in IsomH2. It is a classical result due to C. L.

Siegel [73] that the smallest co-area of a discrete group in IsomH2is achieved

by the Coxeter triangle group [7,3]. Furthermore, one has

area([7,3]) = π−(π

2+π

3+π

7) = π

42 .

In the non-cocompact case, the smallest co-area is realised by the Coxeter

triangle group [∞,3] closely related to the modular group SL2(Z). One has

area([∞,3]) = π−(π

2+π

3) = π

In dimension 3

By a fundamental result of Andreev [2], we dispose of a complete descrip-

tion of Coxeter polyhedra in H3. In fact, the existence of an acute-angled

polyhedron of ﬁnite volume in H3can be deduced from the mutual inter-

section behaviour of its facets and the resulting angular inequalities. As a

consequence, any family of Coxeter polyhedra of ﬁxed combinatorial type in

H3can be listed in detail. In what follows, we give an overview of the most

important families of interest for our work.

All hyperbolic Coxeter tetrahedra have been classiﬁed. This result is mainly

due to Lann´er [54] in the compact case, and to Chein [15] and Koszul [53] in

the non-compact case. They are listed in Figures 3.1.1 and 3.1.2.

In addition, all Coxeter 3-pyramids have been established by Tumarkin [78].

Their Coxeter diagrams are given in Figure 3.1.3. Note that all of them

admit a pair of disjoint facets.

Observe that for all Coxeter polyhedra mentioned above, their dihedral an-

gles are bigger than or equal to π

6. This phenomenon holds for all Coxeter

polyhedra with at most 5 facets in H3; see [23].

Chapter 3. Some classiﬁcation results

5 5 5 5 4 5

4 5

Figure 3.1.1: The compact Coxeter tetrahedra

4 6 4 4 4 4 4

6 6 6 5 6 6

456

4 6

Figure 3.1.2: The non-compact Coxeter tetrahedra

n∞∞

k= 2,3,4 ; m= 2,3,4;

l= 3,4 ; n= 3,4

∞∞

k= 5,6 ; m= 2,3;

l= 2,3,4,5,6

Figure 3.1.3: The Coxeter pyramids in H3

2. In higher dimensions

2 In higher dimensions

In dimensions beyond 3, only a few essentially diﬀerent examples are known.

In this section, we summarize the classiﬁcation results of interest to us.

2.1 Coxeter polyhedra with a small number of facets

The compact Coxeter simplices were classiﬁed by Lann´er, and they exist only

in dimensions n≤4, while non-compact Coxeter simplices were classiﬁed by

Chein [15] and Koszul [53], and exist in dimensions n≤9. The latter sim-

plices are sometimes called quasi-Lann´er. Moreover, all Coxeter polyhedra

in Hnwith precisely n+ 2 facets (respectively, n+ 3 facets in the compact

case) are known due to the works of Esselmann [20, 21], Kaplinskaya [42]

and Tumarkin [78, 79, 80].

This section contains the lists of all Coxeter polyhedra in Hnfor n≥4 with

facet number fn−1=n+ 1, together with classiﬁcation results and sublists

relevant for this thesis when the facet number satisﬁes fn−1≥n+ 2.

The case fn−1=n+ 1

As mention above, compact Coxeter simplices exist for dimensions n≤4

only; see Figure 3.2.4.

5 5 4 5 5

Figure 3.2.4: The compact Coxeter simplices in H4

In the non-compact case, we know that simplices exist up to dimension n= 9.

We give their Coxeter diagrams in Table 3.2.1. Observe that all of them have

dihedral angles π

2,π

3,π

4or π

6, only.

Let us add that the covolumes of all hyperbolic Coxeter simplex groups have

been determined by Johnson, Kellerhals, Ratcliﬀe and Tschantz [40].

The case fn−1=n+ 2

Let now Pbe a Coxeter polyhedron in Hnwith precisely n+ 2 facets. Then,

Pis combinatorially a product of two simplices or a pyramid over a product

of two simplices.

Chapter 3. Some classiﬁcation results

44 4

444

4 4

Table 3.2.1: The non-compact Coxeter simplices in Hnfor n≥4

2. In higher dimensions

Let us ﬁrst consider products of simplices. All simplicial prisms have been

classiﬁed by Kaplinskaya [42] and exist up to dimension n= 5. We emphazise

that all of them contain one pair of disjoint facets.

10 4

5 5

58 4

8 84 4

510

Figure 3.2.5: The Esselmann polyhedra

Assume that Pis combinatorially a product of two simplices each of dimen-

sion greater than 1. If Pis compact, then Pis one of the polyhedra listed in

Figure 3.2.5. They were found by Esselmann [21] and are called Esselmann

polyhedra. There are exactly seven Esselmann polyhedra, and they are all of

dimension 4.

The unique non-compact Coxeter polyhedron being a product of two sim-

plices is the Coxeter polyhedron P0⊂H4found by Tumarkin [78] and de-

picted in Figure 3.2.6.

Figure 3.2.6: The Coxeter polyhedron P0⊂H4

All Coxeter pyramids over a product of two simplices have been classiﬁed by

Tumarkin [78]. They are non-compact and exist in Hnfor dimensions n≤13

and n= 17. In dimension n= 4, all of them contain one pair of parallel

facets, and in particular, their Coxeter diagrams contain a subdiagram of

type e

A1= [∞].

In dimension beyond 4, we emphasize that all of their dihedral angles are

equal to π

2,π

3,π

4or π

6only.

Table 3.2.2 contains all Coxeter pyramids over a product of simplices in Hn

having no pair of disjoint facets and no dihedral angle π

4. These pyramids

exist only for 5 ≤n≤17. They will play a crucial role in Chapter 4.

Chapter 3. Some classiﬁcation results

6 6 6

6 6

Table 3.2.2: The Coxeter pyramids with mutually intersecting facets

and having only dihedral angles of the form π

2,π

3or π

6in Hn

2. In higher dimensions

The case fn−1=n+ 3

For a Coxeter polyhedron P⊂Hnwith n+ 3 facets, the classiﬁcation is

complete in the compact case (see [21, 79]), but it is not complete in the

non-compact ﬁnite volume case. In fact, Coxeter polyhedra with n+ 3 facets

and non-simple vertices are not entirely known; see also [72].

In the compact case, Esselmann [20] showed that Coxeter n-polyhedra with

n+3 facets do not exist in dimensions n > 8, and Tumarkin [80] classiﬁed all

of them. The unique polyhedron in dimension n= 8 is the one found earlier

by Bugaenko [11].

In the non-compact case, Coxeter n-pyramids with n+ 3 facets are pyramids

over a product of three simplices and have been classiﬁed by Tumarkin [79]

as well. They exist for dimensions n≤13. Table 3.2.2 contains the examples

of Coxeter pyramids with mutually intersecting facets and having dihedral

angles π

2,π

3or π

6, only. Observe that they exist only for n= 7.

Another class of non-compact Coxeter polyhedra with n+3 facets of interest

to us are special Napier cycles found by Im Hof [38]. These are doubly-

truncated orthogonal simplices, all of whose dihedral angles are bigger than

or equal to π

Finally, Tumarkin [80] proved that Coxeter polyhedra with n+3 facets do not

exist in dimensions n≥17, and that there exists a unique one in dimension

n= 16.

As for Coxeter polyhedra with at most n+ 3 facets, there are the following

results.

All compact hyperbolic Coxeter polyhedra with at most n+ 3 facets are

classiﬁed; see the above sections.

In [26], Felikson and Tumarkin consider compact Coxeter polyhedra with

ultraparallel facets and proved the following result.

Theorem 3.2.1. A compact Coxeter polyhedron in Hnwith exactly one pair

of ultraparellel facets has at most n+ 3 facets.

As a consequence, all compact hyperbolic Coxeter polyhedra with a unique

pair of disjoint facets are known, and compact Coxeter polyhedra with more

than n+ 3 facets admit at least two pairs of non-intersecting facets.

About fn−1≥n+ 4

Let us ﬁrst mention that, in general, Coxeter n-polyhedra with at least n+4 ≥

8 facets and no further restrictions are not well understood.

Chapter 3. Some classiﬁcation results

One result which we would like to cite here is due to Jacquemet and Tschantz

[39]. They classiﬁed all the Coxeter hypercubes in Hnand showed that they

exist up to dimension n= 5.

A second and important observation concerns the result on pyramids of

Mcleod [61], who completed the classiﬁcation as follows. A pyramid over

a product of more than three simplices is necessarily a pyramid over a prod-

uct of four simplices, and they exist only in dimension 5. Again, each of

them admits a pair of disjoint facets. This fact will be used later on.

For dimensions n= 4 and 5, Coxeter polyhedra with precisely n+ 4 facets

are classiﬁed by works of Burcroﬀ [12] and Ma-Zheng [55, 56]. For dimension

n= 7, there is a unique Coxeter polyhedron with n+ 4 facets, and there are

no such polyhedra for n≥8, as proven by Felikson and Tumarkin in [27].

2.2 Coxeter polyhedra with mutually intersecting facets

In this section, we consider Coxeter polyhedra with mutually intersecting

facets, that is, admitting no pair of disjoint facets in Hn∪∂Hn. In other

words, their Coxeter diagrams have only ﬁnite labels.

In the following, we provide some important results due to Felikson and Tu-

markin concerning the minimal number of pairs of non-intersecting facets

in the subclass of simple Coxeter polyhedra. Then, we discuss the classiﬁ-

cation result due to Prokhorov [70] who considered Coxeter polyhedra with

mutually intersecting facets and dihedral angles π

2and π

3, only.

Some important results

Let P⊂Hnbe a (ﬁnite-volume) Coxeter polyhedron all of whose facets are

mutually intersecting.

The following results due to Felikson and Tumarkin [24] are of fundamental

importance for this thesis.

Theorem 3.2.2. Let P⊂Hnbe a compact Coxeter polyhedron. If all facets

of Pare mutually intersecting, then Pis either a simplex, or it is an Essel-

mann polyhedron.

In the non-compact case, a similar result holds for simple polyhedra; see [24].

Theorem 3.2.3. Let P⊂Hnbe a non-compact simple Coxeter polyhedron.

Assume that all facets of Pare mutually intersecting. Then, Pis either

a simplex, or it is isometric to the polyhedron P0whose Coxeter graph is

depicted in Figure 3.2.7.

2. In higher dimensions

Figure 3.2.7: The Coxeter polyhedron P0⊂H4

As a consequence of Theorem 3.2.2 and Theorem 3.2.3, we deduce the fol-

lowing by-product. In fact, for n≤4, one can drop the simplicity condition

from the combinatorial structure of ideal vertices.

Corollary 3.2.4. Let n≤4. If P⊂Hnis a ﬁnite-volume Coxeter polyhe-

dron with mutually intersecting facets, then Pis a simplex, an Esselmann

polyhedron, or Pis isometric to the polyhedron P0depicted in Figure 3.2.7.

Proof. Let P⊂Hnbe a Coxeter polyhedron with mutually intersecting

facets for 2 ≤n≤4. If Pis simple, by Theorems 3.2.2 and 3.2.3, Pis

either a simplex, an Esselmann polyhedron, or P0, and we are done. Suppose

therefore that Phas a non-simple vertex v∞, that is, v∞is the intersection

of at least n+ 1 facets. As any Coxeter polygon is simple, we can assume

that n > 2.

In the Coxeter diagram Σ of P, the vertex v∞corresponds to an aﬃne Coxeter

subdiagram σ∞of rank n−1 satisfying

rank(σ∞) = order(σ∞)−m, (3.1)

where mis the number of connected aﬃne components of σ∞; see Theorem

1.3.8. Since v∞is non-simple, one has that order(σ∞)≥n+ 1. By means

of (3.1), we deduce that the subdiagram σ∞admits at least m≥2 aﬃne

components. Since n≤4, it follows that σ∞has at least one aﬃne component

of rank 1. More precisely, for n= 3, σ∞=e

A1∪e

A1, and for n= 4, σ∞is

either e

A1∪e

A2,e

A1∪e

G2, or e

A1∪e

A1. Therefore, we derive that Padmits

at least one pair of parallel facets.

By Corollary 3.2.4, all Coxeter polyhedra in Hnwith mutually intersecting

facets are known for n≤4.

The classiﬁcation of ADE-polyhedra

Among all Coxeter polyhedra with mutually intersecting facets, a speciﬁc

subfamily - very important for our research - has been classiﬁed by Prokhorov

in [70]. This family comprises the so-called ADE-polyhedra.

Deﬁnition 3.2.5. An ADE-polyhedron is a Coxeter polyhedron P⊂Hnof

ﬁnite volume satisfying the following properties.

Chapter 3. Some classiﬁcation results

1. All facets of Pare mutually intersecting.

2. All dihedral angles of Pare equal to π

2or π

The terminology is inspired by the spherical and aﬃne cases where Coxeter

polyhedra with mutually intersecting facets and with dihedral angles π

2or π

are of type An,Dnor E6, E7, E8, as well as of type e

An, for n > 1, e

Dn, for

n≥3, or e

E6,e

E7,e

E8; see Tables 1.3.1 and 1.3.2.

By use of the classiﬁcation results for ADE-lattices due to Nikulin [64],

Prokhorov [70] derived that ADE-polyhedra do not exist in Hnfor n > 17,

and he proved the following classiﬁcation result.

Theorem 3.2.6 (Prokhorov [70]).Let Pbe an ADE-polyhedron. Then, P

is either a simplex, a pyramid, or one of the Coxeter polyhedra depicted in

Figure 3.2.8.

Figure 3.2.8: The ADE-polyhedra P1⊂H8and P2⊂H9

In total, there are 34 hyperbolic ADE-polyhedra. The 32 ADE-simplices

and ADE-pyramids can be found in the previous sections; see Figure 3.1.2,

Table 3.2.1 and Table 3.2.2. The highest dimensional ADE-polyhedron is the

Coxeter pyramid in H17 depicted Table 3.2.2.

In the following chapter, we shall extend substantially Prokhorov’s algorithm

and develop further strategies in order to classify so-called ADEG-polyhedra.

These are Coxeter polyhedra with mutually intersecting facets all of whose

dihedral angles are π

2,π

3and (at least one) π

CHAPTER 4

Coxeter polyhedra with mutually intersecting

facets and dihedral angles π

2,π

3and π

In this chapter, we provide the complete classiﬁcation of the following family

of hyperbolic Coxeter polyhedra.

Deﬁnition 4.0.1. A polyhedron P⊂Hnis an ADEG-polyhedron if Pis of

ﬁnite volume and satisﬁes the following properties.

1. All facets of Pare mutually intersecting.

2. All dihedral angles of Pare equal to π

2,π

3or π

3. Phas at least one dihedral angle π

The terminology is inspired by Prokhorov’s work [70] on the classiﬁcation of

ADE-polyhedra.

Let P⊂Hnbe an ADEG-polyhedron and consider the Coxeter group Γ

associated with Pwith its natural presentation given by (1.9), that is,

⟨s1, . . . , sN|s2

i= 1,(sisj)mij = 1⟩.

The ADEG-property of Pimplies that Γ satisﬁes the crystallographic con-

dition which says that mij ∈ {2,3,4,6}for all i, j ∈ {1, . . . , N}; see Chapter

2. The irreducible spherical and aﬃne subgroups of Γ and the related root

systems are hence well understood.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

6 6

Figure 4.0.1: Two Napier cycles in H5

In Chapter 3, we already mentioned some examples of ADEG-polyhedra.

For instance, the two special Napier cycles depicted in Figure 4.0.1 describe

ADEG-polyhedra in H5; see [38].

We state now our main result providing the complete classiﬁcation of ADEG-

polyhedra; see [10].

Theorem 4.0.2. Let P⊂Hnbe an ADEG-polyhedron. Then, Pis one of the

24 Coxeter polyhedra depicted in Table 4.0.1. In particular, Pis non-compact

for n > 2, non-simple for n > 3, and Pis of dimension n≤11. Furthermore,

Pis combinatorially equivalent to one of the following polyhedra.

✧Pis a triangle.

✧Pis a tetrahedron.

✧Pis a doubly-truncated 5-simplex.

✧Pis a pyramid over a product of two or three simplices.

✧Pis the polyhedron P⋆⊂H9with 14 facets depicted in Figure 4.0.2.

Figure 4.0.2: The Coxeter polyhedron P⋆⊂H9

Remark 4.0.3. Apart from the two-dimensional case and one tetrahedron,

all ADEG-polyhedra give rise to Coxeter groups which are arithmetic over

Q; see Chapter 1. The four compact ADEG-triangle groups are arithmetic

as well, but their ﬁeld of deﬁnition is Q(√3); see Example 1.3.18.

6 6

36 6 6 6

6 6

6 6 6

6 6

Table 4.0.1: The ADEG-polyhedra in Hn

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

Remark 4.0.4. The Coxeter polyhedron P⋆⊂H9is a new discovery. It has

14 facets and 134 vertices with 6 of them being ideal. It will turn out that the

group Γ⋆= Γ(P⋆) is commensurable to the group associated with the ADE-

polyhedron P2found by Prokhorov, as well as to all Coxeter simplex groups

and all Coxeter pyramid groups in H9; see Figure 3.2.8 and Proposition 4.3.1.

In particular, we will see that the volumes of all these polyhedra are of the

form q·ζ(5)

22,295,347,200 with q∈Q≥1.

1 The general strategy

The proof of our main result as stated in Theorem 4.0.2 is very involved and

has several ingredients of structural and of quite technical nature. In this

section, we provide a rough overview and the diﬀerent ideas which will show

up. Later on, we will give all details. Some auxiliary data are collected in

the two Appendices A and B.

Let P⊂Hnbe a candidate for an ADEG-polyhedron, and let Σ be its

Coxeter diagram.

❏As Pmust have mutually intersecting facets, we have full control if P

is simple or of dimension n≤4, in view of the results of Felikson and

Tumarkin. Therefore, we will henceforth assume that Pis a non-simple

(and hence non-compact) polyhedron of dimension n≥5.

❏As Pmust have at least one dihedral angle equal to π

6, the diagram

Σ contains a subdiagram σ= [6] of type G2. This property has two

fundamental consequences.

❏A ﬁrst consequence is of combinatorial-metrical nature. By Borcherds’

theorem, σcorresponds to a face F(σ) of codimension 2 in Pwhich is

itself an ADE- or an ADEG-polyhedron. We shall call F(σ) = F(G2)

aG2-face.

❏The second consequence is of algebraic nature. We show that σ= [6]

gives rise to an aﬃne subdiagram [6,3] of type e

G2in Σ.

❏The subdiagram e

G2can be completed to an aﬃne subdiagram σ∞of

rank n−1 in Σ, and σ∞describes a non-simple ideal vertex v∞of P.

❏The vertex v∞appears as the apex of a polyhedral cone C∞⊂Hnover

a product of several simplices, each of dimension ≥2.

2. Proof of the main theorem

❏The cone C∞is of inﬁnite volume and has to be truncated by additional

hyperplanes Hx, x ∈Rn,1, not passing through v∞, in order to yield P.

For two such hyperplanes Hx, Hy, one has ∡(Hx, Hy)∈ {π

2,π

3,π

6}.

❏The ﬁrst hyperplanes Hxwhich we detect have to belong to the bound-

ary of the G2-face F(σ). This set is completed by hyperplanes Hy

subject to the imposed angular condition.

❏The related Lorentzian product ⟨x, y⟩is controlled by Prokhorov’s for-

mula in terms of the (reducible) root lattice corresponding to σ∞. In

this way, one ﬁnds several but ﬁnitely many so-called admissible pairs

of vectors {x, y}.

❏By truncating the cone C∞with all hyperplanes indexed by pairwise ad-

missible vectors, one obtains a polyhedral object P. Finally, it remains

to test if Pis the realisation of a ﬁnite-volume hyperbolic n-polyhedron.

The above procedure is constructive relating Pto its G2-faces, and it yields

all but ﬁnitely many polyhedral candidates in each dimension.

Obviously, by the non-existence result stated in Theorem 3.0.1, the algorithm

will not furnish ADEG-polyhedra of dimensions n≥996. In fact, it stops

much earlier, namely for n= 20.

2 Proof of the main theorem

2.1 First steps

By the results of Felikson and Tumarkin, and by Corollary 3.2.4, all ADEG-

polyhedra in Hnare known for n≤4, and they are all simple. In fact, all

simple ADEG-polyhedra are known in any dimension, and they are given by

certain Coxeter simplices. They exist up to dimension 3. More precisely, in

dimension 2, there are four compact triangles, and in dimension 3, there are

seven non-compact tetrahedra, all depicted in Table 3.2.1.

Moreover, all Coxeter pyramids are classiﬁed, and we can extract all ten

ADEG-pyramids from Table 3.2.2.

In the sequel we will consider ADEG-polyhedra of dimensions n≥5, only.

They are non-simple, and we can suppose that they are diﬀerent from (the

known) Coxeter pyramids. We point out that all of their G2-faces are non-

compact.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

2.2 Existence of a component e

Let Pbe an ADEG-polyhedron in Hnfor n≥5. Let Σ be its Coxeter

diagram. By hypothesis, it contains a subdiagram σ= [6] of type G2in Σ.

We show that Σ contains in fact a subdiagram [6,3] of type e

G2. This result

will be essential.

Lemma 4.2.1. Let n≥5, and let P⊂Hnbe an ADEG-polyhedron. Then,

the Coxeter diagram of Pcontains at least one subdiagram of type e

G2.

Proof. Let n≥5, and let P⊂Hnbe an ADEG-polyhedron. Denote by Σ its

Coxeter diagram. Since Pis an ADEG-polyhedron, Σ contains a subdiagram

σ= [6].

By Borcherds’ result stated in Theorem 1.3.9, σyields a G2-face F=F(G2)

of dimension n−2 which is an ADE- or an ADEG-polyhedron. As n−2≥3,

Fis non-compact. Denote by σFthe Coxeter diagram of F, and notice that

σFis disjoint from σ.

Now, σFcontains an aﬃne subdiagram σ∞

Fof rank n−3 which appears as

a component in an aﬃne diagram of rank n−1 in Σ. As the complement

Σ\σFcontains σ, the aﬃne subdiagram of Σ\σ∞

Fcoincides with the diagram

[6,3] of type e

G2.

2.3 From a G2-face to an admissible set of vectors

Let P⊂Hnbe an ADEG-polyhedron for n≥5, and let Σ be its Coxeter

diagram.

By Lemma 4.2.1, there is an aﬃne subdiagram in Σ of the form

σ∞=σ1∪σ2∪ ··· ∪ σm=e

G2∪σ2∪ ··· ∪ σm, m ≥2.

As before, denote by rithe rank of σiso that Pm

i=1 ri=n−1.

In our setting, each aﬃne component σican be interpreted as the extended

Dynkin diagram of a root system Riof type A, D, E or G2; see Chapter 2.

Denote by ei

1, . . . , ei

rithe natural simple roots (with their prescribed lengths),

together with ei

ri+1 given by the opposite of the highest root of Ri.

The diagram σ∞corresponds to a non-simple ideal vertex v∞of the poly-

hedron P. The vectors ei

jassociated with σican be interpreted as spacelike

vectors normal to hyperplanes Hei

jpassing through v∞. Furthermore, we

treat v∞as a lightlike vector in Rn,1. In particular, v∞is Lorentz-orthogonal

2. Proof of the main theorem

to all ei

j, and it can be expressed in terms of certain lightlike vectors vi

∞

related to the aﬃne components σifor i= 1,...m as follows.

Lemma 4.2.2. For each 1≤i≤m, there exist positive integers ci

1, . . . , ci

ri+1

with ci

ri+1 = 1 such that the vector vi

∞:= ci

1ei

1+··· +ci

ri+1ei

ri+1 ∈Rn,1is

collinear with v∞.

Proof. Let ci

1, . . . , ci

ri∈Zbe the integers such that

−eri+1 =ci

1ei

1+. . . +ci

riei

ri∈Rri; (4.1)

see Chapter 2 for their explicit form. The vector vi

∞=ci

1ei

1+. . .+ci

riei

ri+ei

ri+1

(suitably extended to a non-zero vector in Rn,1) satisﬁes ⟨vi

∞, vi

∞⟩= 0 by

properties of the highest root. In addition, ⟨vi

∞, v∞⟩= 0 as ⟨ei

j, v∞⟩= 0 for

all j= 1, . . . , ri+ 1. Now, we conclude by means of Lemma 1.2.1.

As a consequence of Lemma 4.2.2, and for each i, the vectors v∞and vi

∞

are collinear and deﬁne the same lightlike ray, symbolized by v∞∼vi

∞. In

particular, for i= 1 and σ1=e

G2, we derive that

v∞∼v1

∞= 3e1

1+ 2e1

2+e1

3; (4.2)

see Deﬁnition 2.3.6.

Let us come back to the set of hyperplanes Hei

jwith 1 ≤j≤ri+ 1 and

1≤i≤m. The goal is to complete this set by at least two additional

hyperplanes Hx, Hy, where x, y ∈Rn,1are spacelike vectors, in order to form

the boundary of an ADEG-polyhedron.

In what follows, we will always consider xand yto have squared norm 2.

We characterize the vectors x, y ∈Rn,1in terms of their Lorentzian products

j:= ⟨x, ej

i⟩and li

j:= ⟨y, ej

i⟩with the vectors ei

More precisely, we encode xand yby strings in the following way.

x↔(k1

1, k1

2, k1

3;k2

1, . . . , k2

r2+1;. . . ;km

1, . . . , km

rm+1)

y↔(l1

1, l1

2, l1

3;l2

1, . . . , l2

r2+1;. . . ;lm

1, . . . , lm

rm+1)

(4.3)

Since ∡(Hx, Hei

j) and ∡(Hy, Hei

j) vary within the set {π

2,π

3,π

6}, one has

j, li

j∈ {0,−1,−√3,−3}. More speciﬁcally, the quantities ki

j=⟨x, ei

j⟩are

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

as follows.

⟨x, ei

j⟩=











0 if ∡(Hx, Hei

j) = π

−1 if ∡(Hx, Hei

j) = π

3and ||ei

j||2= 2

−√3 if ∡(Hx, Hei

j) = π

6and ||ei

j||2= 2

−√3 if ∡(Hx, Hei

j) = π

3and ||ei

j||2= 6

−3 if ∡(Hx, Hei

j) = π

6and ||ei

j||2= 6

(4.4)

Similar expressions hold for li

j=⟨y, ei

j⟩.

In this setting, the Lorentzian product ⟨x, y⟩can be expressed in terms of

the string coeﬃcients of xand ygiven by (4.3). This point of view is taken

from work of Nikulin [65] and Prokhorov [70], and we provide the technical

details in what follows.

Fix two spacelike vectors x, y ∈Rn,1such that ⟨x, v∞⟩ = 0 ,⟨y, v∞⟩ = 0, that

is, the hyperplanes Hxand Hydo not pass through the vertex v∞.

Consider now the quantities Λ and Λi, 1 ≤i≤m, deﬁned as follows.

Λ=Λxy := ⟨x, v∞⟩

⟨y, v∞⟩and Λi:= ⟨x, vi

∞⟩

⟨y, vi

∞⟩=⟨x, Pri+1

k=1 ci

kei

k⟩

⟨y, Pri+1

k=1 ci

kei

k⟩,

where the lightlike vectors vi

∞are deﬁned in Lemma 4.2.2.

In particular, in our setting where σ1=e

G2, one has

Λ1=3k1

1+ 2k1

2+k1

3l1

1+ 2l1

2+l1

.(4.5)

Since vi

∞∼v∞by Lemma 4.2.2, one immediately deduces that

Λi= Λ for i= 1, . . . , m . (4.6)

With these preparations, we are able to write down Prokhorov’s formula [70]

for the Lorentzian product ⟨x, y⟩.

Prokhorov’s formula.

⟨x, y⟩=⟨x, x⟩

2Λ +⟨y, y⟩

2Λ−(∆1+. . . + ∆m),(4.7)

with

∆p=X

1≤i<j≤rp+1

(kp

ilp

j+kp

jlp

i−kp

ikp

Λ−lp

ilp

jΛ)sp

ij .(4.8)

2. Proof of the main theorem

The quantities sp

ij in (4.8) depend on the fundamental weights of the root

system Rpas follows; see [70]. Let w1, . . . , wrpbe the fundamental weights

of the root system Rpas described in Chapter 2, and let cp

jbe the integers

according to (4.1). Then,











ij =⟨wi, wj⟩ − ⟨wi, wi⟩cp

2cp

i−⟨wj, wj⟩cp

2cp

for 1 ≤i, j ≤rp,

irp=⟨wi, wi⟩

2cp

for i= 1, . . . , rp,

ii = 0 for i= 1, . . . , rp+ 1 .

(4.9)

In the following example we give the explicit form of these quantities for a

component of type e

G2. For the aﬃne components of type e

A, e

Dor e

E, we refer

to Appendix A.

Example 4.2.3. For the root system G2underlying e

G2, the roots indexing

the nodes of the diagram e

G2, as depicted below, are such that ||e1||2= 2,

and ||e2||2=||e3||2= 6; see Deﬁnition 2.3.6. One has ⟨e1, e2⟩=⟨e2, e3⟩=−3

and ⟨e1, e3⟩= 0.

e1e2e3

Furthermore, c1= 3, c2= 2 and c3= 1, and the fundamental weights for G2

are given by w1= 2e1+e2

w2= 3e1+ 2e2.

Hence ⟨w1, w1⟩= 2,⟨w2, w2⟩= 6, ⟨w1, w2⟩= 3. From (4.9), it follows that

s12 =13

6, s23 =1

3and s13 =3

Let us return to the non-simple ideal vertex v∞of the ADEG-polyhedron

P⊂Hnand its associated aﬃne diagram

σ∞=e

G2∪σ2∪ ··· ∪ σm, m ≥2.

In this context, consider two spacelike vectors x, y of squared norm 2 such

that ⟨x, v∞⟩ = 0 ,⟨y, v∞⟩ = 0, together with their strings (4.3).

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

Deﬁnition 4.2.4. The vectors x, y form an admissible pair, denoted by

{x, y}, if they fulﬁl the following conditions.

Λ1=3k1

1+ 2k1

2+k1

3l1

1+ 2l1

2+l1

=Pjc2

jk2

Pjc2

jl2

=··· =Pjcm

jkm

Pjcm

jlm

= Λm= Λ ,(4.10)

and

⟨x, y⟩= Λ + 1

Λ−(∆1+. . . + ∆m)∈ {0,−1,−√3},(4.11)

where ∆1,...,∆mare given by (4.8).

A set of vectors {z1, . . . , zt}is said to be admissible if its elements are pairwise

admissible, that is, {zi, zj}is an admissible pair for all i, j ∈ {1, . . . , t},i=j.

Now, consider the subdiagram σ= [6] of the component e

G2in σ∞, and let

F=F(σ) be the G2-face with Coxeter diagram σF. By Borcherds’ result,

σFcontains the aﬃne diagram σ2∪ ··· ∪ σm, and it is hyperbolic.

Denote by x1, . . . , xtall the nodes in σF\(σ2∪ ··· ∪ σm). They represent

hyperplanes Hx1, . . . , Hxtnot containing v∞, and we take them of squared

norm 2. Furthermore, they are good neighbours of σimplying that

⟨xi, e1

1⟩=⟨xi, e1

2⟩= 0 for i= 1, . . . , t .

To begin with, suppose that t= 1, and write x=x1. Since xis a good

neighbour of σ, its string can be written as

x↔(0,0, a;k2

1,...k2

r2+1;. . . ;km

1, . . . , km

rm+1),(4.12)

where we abbreviate a:= k1

3=⟨x, e1

3⟩∈{0,−√3,−3}; see (4.4). Observe

that a= 0, since otherwise the complement of σFin Σ is not equal to σ.

By assumption, the ADEG-polyhedron Pis not a pyramid; see Section 2.1.

Therefore, there is at least one additional hyperplane Hythat is not passing

through v∞. The corresponding node is a bad neighbour of σ, and obviously

outside of σF.

Suppose that yis of squared norm 2, and put

y↔(l1

1, l1

2, l1

3;l2

1, . . . , l2

r2+1;. . . ;lm

1, . . . , lm

rm+1)

for its string. As yis a bad neighbour of σ,l1

1and l1

2are not simultaneously

zero.

Notation. In what follows, for simplicity, we replace ki

jby its opposite and

write ki

j=−⟨x, ei

j⟩from now on. Similarly, we denote li

j=−⟨y, ei

j⟩.

2. Proof of the main theorem

Since Pshould be an ADEG-polyhedron, the vectors xand yform an admissi-

ble pair {x, y}. If the hyperplanes Hxand Hycombined with the hyperplanes

Hei

j,1≤j≤ri+ 1,1≤i≤m, bound a ﬁnite-volume polyhedron in Hn,

we realised an ADEG-polyhedron whose combinatorial-metrical structure is

explicit. Otherwise, for t= 1, we have to search for additional vectors yi

such that {x, yi}and {yi, yj},i=j, are admissible pairs.

Next, suppose that t≥2. The vectors x1, . . . , xtare all encoded by strings

according to (4.12) where

ai:= −⟨xi, e1

3⟩∈{√3,3}.(4.13)

Since Pis an ADEG-polyhedron, the set {x1, . . . , xt}is admissible. As above,

either the hyperplanes Hxi, together with the hyperplanes Hei

j, bound a

ﬁnite-volume polyhedron in Hn, or one has to add vectors yito {x1, . . . , xt}

to produce an admissible set.

Remark 4.2.5. Each G2-face F⊂Pis an ADE- or ADEG-polyhedron

of dimension n−2≥3. By ﬁxing one such polyhedron, we ﬁx the string

coeﬃcients of the corresponding vectors x1, . . . , xtup to the terms given by

(4.13). In this way, and by comparing with Prokhorov’s way, our classiﬁcation

procedure in the ADEG-case is more eﬃcient.

2.4 The classiﬁcation of ADEG-polyhedra

Let n≥5, and let P⊂Hnbe an ADEG-polyhedron with Coxeter diagram

Σ. Assume that Pis not a pyramid. As in the previous sections, denote by

v∞a non-simple vertex of Pwhose associated subdiagram is of the form

σ∞=e

G2∪σ2∪ ··· ∪ σm, m ≥2.

Recall that the G2-face F=F(σ)⊂Hn−2, where σ= [6] ⊂e

G2, is a non-

compact ADE- or ADEG-polyhedron. Denote by σFits Coxeter diagram.

We work inductively on the dimension n≥5 by taking into account the

knowledge of all ADE- and ADEG-polyhedra of dimensions 3 and 4.

This induction process ﬁnishes in dimension nif there exists neither an ADE-

polyhedron nor an ADEG-polyhedra in Hn−2which could serve as a G2-face

of an ADEG-polyhedron in Hn. A priori, the procedure has to be performed

at least up to dimension n= 19, as there exists one ADE-pyramid in H17.

In practice, we ﬁx a face G2-face F, and denote by x1, . . . , xtthe nodes which,

together with σ2∪ ··· ∪ σm, constitute the diagram σF.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

Observe that in the case where Fis a simplex or a pyramid, one necessarily

has t= 1, and one has t≥2 otherwise.

By means of Prokhorov’s formula, and with the help of the software Mathe-

matica [85], we ﬁnd all the vectors ywhich together with the vectors corre-

sponding to x1, . . . , xtform an admissible set; see Section 2.3.

Then, for each admissible set, we verify if the arrangement of the diﬀerent hy-

perplanes of the form Hei

j,Hxand Hygive rise to a ﬁnite-volume polyhedron

in Hn. To this end, we apply one of the following three criteria.

✧The signature of the Gram matrix is (n, 1).

✧Vinberg’s ﬁnite-volume criterion stated in Theorem 1.3.15.

✧Each (n−2)-face has to be of ﬁnite volume.

For example, spherical subdiagrams of rank > n cannot appear, and any

aﬃne subdiagram is a component of an aﬃne subdiagram of rank n−1.

We are now ready to start the induction procedure.

Assume that n =5

By Lemma 4.2.1, the aﬃne diagram σ∞=σ1∪σ2of rank 4 is either e

G2∪e

or e

G2∪e

A2. In addition, we know that the 3-face F=F(G2) is a ﬁnite-volume

Coxeter polyhedron whose Coxeter diagram σFcontains σ2. Therefore, Fis

an ADE- or ADEG-tetrahedron, and there are ten possibilities for σF; see

Figure 3.1.2.

Consider the vector x∈R5,1,||x||2= 2, that together with e2

1, e2

2and e2

forms σF. For the string of x, we have

x↔(0,0, a;k2

1, k2

2, k2

3) (4.14)

where a=−⟨x, e1

3⟩∈{√3,3}and k2

j=−⟨x, e2

i⟩∈{0,1,√3,3}.

Table 4.2.2 contains all relevant strings for xto be considered, written as

in (4.14). In fact, we omit the ones that lead to symmetric conﬁgurations.

Notice also that in some cases the coeﬃcient a=√3 for xcan be excluded.

The reason will become clear in part (i) below; see Remark 4.2.6.

⋄Suppose that σ∞=e

G2∪e

G2. Then, Fis one of the seven ADEG-

tetrahedra; see Figure 3.1.2. In what follows, we describe the two most

important cases, only. For all other cases, we give just a few details, but

display all the admissible pairs in Table 4.2.3.

2. Proof of the main theorem

σ∞String for x σ∞String for x

G2∪e

G2(0,0,3; 0,0,√3) e

G2∪e

A2(0,0,3; 1,0,0)

(0,0, a; 1,0,0) (0,0, a; 1,0,1)

(0,0,3; 0,0,3) (0,0, a; 1,1,1)

(0,0, a; 0,√3,0) (0,0, a;√3,0,0)

(0,0, a; 1,0,√3)

(0,0, a; 1,0,3)

Table 4.2.2: Strings for dimension n= 5

(i) Assume that Fis the Coxeter tetrahedron with Coxeter symbol [3,3,6].

The situation can be described with the following diagram.

1e1

2e1

3x e2

3e2

2e2

6 6

We get x↔(0,0, a; 0,0,√3) for a∈ {√3,3}.

A vector y↔(l1

1, l1

2, l1

3;l2

1, l2

3), ||y||2= 2, forms an admissible pair with xif

Λ1=a

3l1

1+ 2l1

2+l1

=√3

3l2

1+ 3l2

2+l2

= Λ2= Λ ,

and if ⟨x, y⟩= Λ + 1

Λ−(∆1+ ∆2)∈ {0,−1,−√3}with ∆1and ∆2as in

(4.8).

Remark 4.2.6. For a=√3, the Coxeter diagram of a 5-pyramid of ﬁnite-

volume appears as a proper subdiagram of Σ which is impossible in view of

Proposition 1.3.17.

As a consequence of Remark 4.2.6, we can assume that a= 3. There is a

unique solution so that {x, y}is admissible, and it is given by

y↔(l1

1, l1

2, l1

3;l2

1, l2

2, l2

3)=(√3,0,0; 1,0,0)

for which Λ = 1

√3, ∆1=√3 and ∆2=1

√3, and one gets ⟨x, y⟩= 0.

This conﬁguration corresponds to the Napier cycle depicted below wherein

the red nodes correspond to the vectors xand y.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

6 6

(ii) Assume that Fis the tetrahedron with Coxeter symbol [3,6,3]. The

situation can be described with the following diagram.

1e1

2e1

3e2

2e2

6 6

For x↔(0,0,√3; 1,0,0), there is unique vector ysuch that the pair {x, y}

is admissible. It is given by y↔(0,√3,√3; 1,0,0) and yields ⟨x, y⟩= 0.

This conﬁguration, depicted below, does not correspond to a ﬁnite-volume

5-polyhedron.

6 6

In fact, this can be easily deduced from the Coxeter diagram as it contains

an aﬃne subdiagram e

A2which cannot be completed to yield an aﬃne sub-

diagram of rank 4. This contradicts Theorem 1.3.15.

For x↔(0,0,3; 1,0,0), there is again a unique solution y↔(1,0,0; 0,0,3)

which satisﬁes ⟨x, y⟩= 0. This gives rise to the Napier cycle depicted below.

6 6

For all of the remaining cases, the admissible pairs are listed in Table 4.2.3.

However, none of them gives rise to a 5-polyhedron of ﬁnite volume. Fur-

thermore, there exists no admissible set of cardinality >2.

⋄In the case where σ∞=e

G2∪e

A2, none of the admissible pairs yields a

ﬁnite-volume polyhedron. In addition, we ﬁnd one admissible set formed by

three vectors. The set of their strings is as follows.

{(0,0,3; 1,0,0),(√3,0,0; 0,0,√3),(0,0,3; √3,0,0)}.

However, also in this case, we do not obtain a ﬁnite-volume polyhedron.

This ﬁnishes the proof for n= 5.

2. Proof of the main theorem

σ∞x y ⟨x, y⟩

G2∪e

G2(0,0,3; 0,0,√3) (√3,0,0; 1,0,0) 0

(0,0,√3; 1,0,0) (1,0,0; 0,√3,√3) 0

(0,0,3; 1,0,0) (1,0,0; 0,0,3) 0

(0,0,3; 0,0,3) (1,0,0; 1,0,0) 0

(0,0,3; 0,√3,√3) (√3,0,0; 1,3,0) 0

G2∪e

A2(0,0,3; 1,0,0) (√3,0,0; 0,0,√3) 0

(0,0,3; √3,0,0) (√3,0,0; 0,0,√3) 0

(0,0,3; √3,0,0) (√3,0,0; 0,√3,0) 0

Table 4.2.3: Admissible pairs {x, y}for n= 5

Assume that n =6

By means of Lemma 4.2.1, there is a unique case to consider given by σ∞=

G2∪e

A3. Hence, there are two possibilities for F, since σFhas to contain

to component e

A3. More precisely, σFis one of the two corresponding ADE-

simplices; see Table 3.2.1.

By Remark 4.2.6, we only have to consider the strings x↔(0,0,3; 1,0,0,0)

and x↔(0,0, a; 1,0,1,0) for a∈ {√3,3}. In both cases, we ﬁnd that there

is no admissible pair.

Hence, apart from a single pyramid, there are no further ADEG-polyhedra

in H6.

Assume that n =7

There are eight possibilities for the diagram σF. Namely, Fis one of the

three ADE-simplices, the three ADEG-pyramids and the two Napier cycles

in H5given in Figure 4.0.1.

If Fis a simplex or a pyramid, there is only one vector xto be added to the

set {ei

j}i,j in order to form σF. If Fis a Napier cycle, two vectors x1and x2

are added to {ei

j}i,j to form σF. By taking into account Remark 4.2.6, we

list all corresponding strings in Table 4.2.4.

In what follows, we only give the details for the case where σ∞=e

G2∪e

G2.

Hence, Fis the pyramid with the Coxeter symbol [6,3,3,3,3,6], or one of

the two Napier cycles.

For all the other cases, the admissible pairs are summarized in Table 4.2.5.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

σ∞Strings for x

G2∪e

A4(0,0,3;1,0,0,0,0)

G2∪e

D4(0,0,3;1,0,0,0,0)

(0,0, a; 0,1,0,0,0)

G2∪e

A2∪e

A2(0,0,3;1,0,0;1,0,0)

G2∪e

A2(0,0,3; 0,0,√3; 1,0,0)

G2∪e

G2(0,0,3; 0,0,√3; 0,0,√3)

{x1↔(0,0, a1; 0,0,3; 1,0,0),

x2↔(0,0, a2; 1,0,0; √3,0,0)}

{x1↔(0,0, a1; 0,0,√3; 0,0,3),

x2↔(0,0, a2; 0,0,3; 1,0,0)}

Table 4.2.4: Strings for dimension n= 7

(i) Assume that σF= [6,3,3,3,3,6]. By Remark 4.2.6, we consider the vector

xwith the string

x↔(0,0,3; 0,0,√3; 0,0,√3) .

There are two vectors yisuch that {x, yi}is an admissible pair, namely

y1↔(√3,0,0; 0,0,3; 1,0,0) , y2↔(√3,0,0; 1,0,0; 0,0,3),

and one has ⟨x, y1⟩=⟨x, y2⟩= 0. Moreover, we verify that {y1, y2}is also an

admissible pair. However, there is no admissible set giving rise to a ﬁnite-

volume 7-polyhedron.

(ii) Assume that Fis the Napier cycle depicted below.

In this case, we can choose

x1↔(0,0, a1; 0,0,√3; 0,0,3) and x2↔(0,0, a2; 1,0,0; √3,0,0) .

Then, we determine the coeﬃcients a1, a2∈ {√3,3}so that {x1, x2}is an

admissible pair. As Λ1=a1

a2and Λ2= Λ3=1

√3, this yields a1=a2

√3, that

is, a1=√3 and a2= 3. It follows that ⟨x1, x2⟩= 0. However, the resulting

polyhedron does not have ﬁnite volume.

2. Proof of the main theorem

Next, we search for all vectors ysuch that {x1, x2, y}is an admissible set.

The only solution is given by y↔(1,0,0; 0,0,3; √3,0,0), and again, we do

not get a ﬁnite-volume polyhedron.

(iii) Assume that Fis the Napier cycle given below.

6 6

We write x1↔(0,0, a1; 1,0,0; 0,0,3) and x2↔(0,0, a2; 0,0,3; 1,0,0).

As above, since Λ2= Λ3= 1, this amounts to a1=a2∈ {√3,3}. As

a consequence, ∆1= 0, and for each a:= a1=a2∈ {√3,3},{x1, x2}is

admissible and such that ⟨x1, x2⟩= 0. None of the resulting conﬁgurations

yields a ﬁnite-volume polyhedron.

For a=√3, we ﬁnd one vector y↔(1,0,0; √3,0,0; √3,0,0) such that

{x1, x2, y}is admissible. Also, this conﬁguration does not give rise to a

ﬁnite-volume polyhedron.

For a= 3, we ﬁnd all the vectors zisuch that {x1, x2, zi}is admissible. They

are given by

z1↔(1,0,0; 0,0,3; 0,0,3) , z2↔(1,0,0; 1,0,0; 1,0,0) ,

z3↔(√3,0,√3;√3,0,√3; √3,0,√3) .

In addition, {z1, z2}and {z1, z3}are admissible pairs. However, for each of

the diﬀerent admissible sets, we do not obtain a ﬁnite-volume polyhedron.

σ∞x y ⟨x, y⟩

G2∪e

D4(0,0,√3; 0,1,0,0,0) (0,√3,√3; 1,1,1,1,1) 0

G2∪e

A2∪e

A2(0,0,3;1,0,0;1,0,0) y1↔(√3,0,0; 0,0,√3; √3,0,0) 0

y2↔(√3,0,0; 0,√3,0; √3,0,0) 0

G2∪e

A2(0,0,3; 0,0,√3; 1,0,0) (√3,0,0; 1,0,0; √3,0,0) 0

Table 4.2.5: The remaining admissible pairs {x, y}for n= 7

For the remaining cases, we list all admissible pairs in Table 4.2.5. Notice

that there is no admissible pair for σ∞=e

G2∪e

A4.

The pair {y1, y2}as given in Table 4.2.5 for σ∞=e

G2∪e

A2∪e

A2is not admis-

sible. Again, none of the admissible pairs yields a ﬁnite-volume polyhedron

in H7.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

Assume that n =8

By means of Lemma 4.2.1, σFis either one of the two ADE-simplices or one

of two ADEG-pyramids. The corresponding strings are listed in Table 4.2.6.

σ∞String for x

G2∪e

A5(0,0, a; 1,0,0,0,0,0)

G2∪e

D5(0,0, a; 1,0,0,0,0,0)

G2∪e

A2∪e

A3(0,0, a; 1,0,0; 1; 0,0,0)

G2∪e

A3(0,0, a; 0,0,√3; 1; 0,0,0)

Table 4.2.6: Strings for dimension n= 8

Let us give some details for the case where σ∞=e

G2∪e

A5, only. By Table 4.2.7,

there are six admissible pairs of the form {x, yi}. We determine which of the

vectors y1, . . . , y6form admissible pairs and give their Lorentzian products

⟨yi, yj⟩in Table 4.2.8. Empty boxes correspond to inadmissible pairs.

σ∞x y ⟨x, y⟩

G2∪e

A5(0,0,√3; 1,0,0,0,0,0) y1↔(0,√3,√3; 0,0,0,1,1,1) -1

y2↔(0,√3,√3; 0,1,1,1,0,0) 0

y3↔(0,√3,√3; 1,1,0,0,0,1) 0

y4↔(1,3,3; √3,0,0,√3,√3,√3) 0

y5↔(1,3,3; √3,√3,√3,√3,0,0) 0

y6↔(√3,√3,0; 1,1,1,0,0,1) 0

G2∪e

D5(0,0,√3; 1,0,0,0,0,0) (0,√3,√3; 1,1,0,0,0,0) 0

(√3,√3,0; 1,1,0,1,1,0) 0

G2∪e

A2∪e

A3(0,0,3; 1,0,0; 1; 0,0,0) (√3,0,0; 0,0,√3; √3,0,0,0) 0

(√3,0,0; 0,√3,0; √3,0,0,0) 0

G2∪e

A3(0,0,√3; 0,0,√3; 1; 0,0,0) (0,√3,√3; √3,0,0; 1,1,0,1) -1

(√3,0,0; 0,√3,√3; 1,1,0,1) -1

(0,0,3; 0,0,√3; 1; 0,0,0) (√3,0,0; 1,0,0; √3,0,0,0) 0

Table 4.2.7: Admissible pairs {x, y}for n= 8

By looking at each admissible set, we see that that none of them gives rise

to a ﬁnite-volume polyhedron in H8.

2. Proof of the main theorem

y2y3y4y5y6

y10 0 −√30

y20−√30

y30 0

y40

y50

Table 4.2.8: The Lorentzian products ⟨yi, yj⟩

We proceed similarly for the other diagrams σ∞, with the diﬀerence that, in

these cases, there are no admissible sets of cardinality >2. The conclusion

remains the same, and the proof for n= 8 is ﬁnished.

Interlude. For n≥9, the number of admissible pairs increases and ad-

missible sets of big cardinality show up. Despite the considerable amount of

cases to test, only a single one will ﬁnally provide a realisation of a hyperbolic

polyhedron of ﬁnite volume. As announced in Theorem 4.0.2, it will be the

polyhedron P⋆⊂H9.

In view of all the lengthy work performed, we will just summarize our ﬁnd-

ings, apart from a few cases, in Appendix B. The cases which we explain in

some more details are the ones where the G2-face F⊂Hn−2, for n= 9, is

neither a simplex nor a pyramid, and where F⊂H7is the pyramid with

apex v∞of type e

G2∪e

G2.

In Appendix B, we list all admissible pairs {x, yi}for a given σ∞, together

with the Lorentzian products ⟨x, yj⟩, as well as the products ⟨yi, yj⟩when

the pair {yi, yj}is admissible and of further relevance; see Table 4.2.8, for

example.

Assume that n =9

By means of Lemma 4.2.1, there are twelve possibilities for F. Namely, there

are three simplices, ﬁve pyramids over a product of two simplices, and four

pyramids over a product of three simplices; see Tables 3.2.1 and 3.2.2. The

corresponding strings are listed in Table 4.2.9; see Remark 4.2.6.

Remarquable is the case where σ∞=e

G2∪e

G2, and where Fis the

7-pyramid over a product of three simplices depicted below.

6 6

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

σ∞String for x

G2∪e

A6(0,0,√3; 1,0,0,0,0,0,0)

G2∪e

D6(0,0, a; 1,0,0,0,0,0,0)

G2∪e

E6(0,0, a; 1,0,0,0,0,0,0)

G2∪e

A3∪e

A3(0,0, a; 1,0,0,0; 1,0,0,0)

G2∪e

A2∪e

A4(0,0, a; 1,0,0; 1,0,0,0)

G2∪e

A4(0,0, a; 0,0,√3; 1,0,0,0,0)

G2∪e

A2∪e

D4(0,0, a; 1,0,0; 1,0,0,0)

G2∪e

D4(0,0, a; 0,0,√3; 1,0,0,0,0)

G2∪e

A2∪e

A2(0,0, a; 0,0,1; 0,0,1; 0,0,1)

G2∪e

A2∪e

A2(0,0, a; 0,0,√3; 0,0,1; 0,0,1)

G2∪e

A2(0,0, a; 0,0,√3; 0,0,√3; 1,0,0)

G2∪e

G2(0,0, a; 0,0,√3; 0,0,√3; 0,0,√3)

Table 4.2.9: Strings for dimension n= 9

As before, we denote x↔(0,0, a; 0,0,√3; 0,0,√3; 0,0,√3) for a∈ {√3,3}.

By applying Borcherds’ theorem to one of the subdiagrams [6] in (the sym-

metric) diagram σF, one immediately deduces that a=√3.

We ﬁnd that there is a unique and beautiful admissible pair {x, y}with y

given by

y↔(1,0,0; 1,0,0; 1,0,0; 1,0,0) ,

and we have ⟨x, y⟩= 0. The corresponding diagram is depicted below.

By means of Theorem 1.3.15, we verify that this polyhedron is a Coxeter

polyhedron of ﬁnite volume in H9! We call it P⋆. For more details and

further results about P⋆, see Section 3.1.

For all other cases, see Tables 2.0.1, 2.0.2, 2.0.5, 2.0.6, 2.0.7, 2.0.3, 2.0.8,

2.0.9, 2.0.10, 2.0.11 and 2.0.12 in Appendix B.

Assume that n =10

There are seven possibilities for F. Namely, there are three ADE-simplices,

three ADEG-pyramids and the ADE-polyhedron P1of Prokhorov in H8.

2. Proof of the main theorem

All corresponding strings are listed in Table 4.2.10.

σ∞String for x

G2∪e

A7(0,0, a; 1,0,0,0,0,0,0,0)

{x1↔(0,0, a1; 1,1,0,0,0,0,0,0),

x2↔(0,0, a2; 0,0,1,1,0,0,0,0),

x3↔(0,0, a3; 0,0,0,0,1,1,0,0),

x4↔(0,0, a4; 0,0,0,0,1,1,0,0)}

G2∪e

D7(0,0, a; 1,0,0,0,0,0,0,0)

G2∪e

E7(0,0, a; 1,0,0,0,0,0,0,0)

G2∪e

A3∪e

D4(0,0, a; 1,0,0,0; 1,0,0,0,0)

G2∪e

A2∪e

D5(0,0, a; 1,0,0; 1,0,0,0,0)

G2∪e

D5(0,0, a; 0,0,√3; 1,0,0,0,0)

Table 4.2.10: Strings for dimension n= 10

Consider the case σ∞=e

G2∪e

A7. When the face Fis an ADE-simplex, we

refer to Table 2.0.13 in Appendix B.

Of interest here is the case where the face Fis given by Prokhorov’s polyhe-

dron P1depicted below.

Let us denote

x1↔(0,0, a1; 1,1,0,0,0,0,0,0) , x2↔(0,0, a2; 0,0,1,1,0,0,0,0) ,

x3↔(0,0, a3; 0,0,0,0,1,1,0,0) , x4↔(0,0, a4; 0,0,0,0,0,0,1,1) ,

where ai∈ {√3,3}for i= 1,2,3,4. It is easy to see that the set {x1, x2, x3, x4}

is admissible if and only if a1=a2=a3=a4. Let a:= ai, 1 ≤i≤4. In both

cases, for a=√3 and a= 3, the corresponding diagram does not encode a

ﬁnite-volume polyhedron. Furthermore, we do not ﬁnd any additional vector

ysuch that the set {x1, x2, x3, x4, y}is admissible.

For all other cases, see Tables 2.0.14, 2.0.15, 2.0.16, 2.0.17 and 2.0.18 in

Appendix B.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

Assume that n =11

There are eight possibilities for F. Namely, there are three ADE-simplices,

three ADEG-pyramids, the ADE-polyhedron P2of Prokhorov and the ADEG-

polyhedron P⋆in H9. The corresponding strings are listed in Table 4.2.11.

We provide details only for F=P2and F=P⋆.

σ∞String for x

G2∪e

A8(0,0, a; 1,0,0,0,0,0,0,0,0)

{x1↔(0,0, a1; 1,0,1,0,0,0,0,0,0),

x2↔(0,0, a2; 0,0,0,1,0,1,0,0,0),

x3↔(0,0, a3; 0,0,0,0,0,0,1,0,1)},

G2∪e

D8(0,0, a; 1,0,0,0,0,0,0,0,0)

G2∪e

E8(0,0, a; 1,0,0,0,0,0,0,0,0)

G2∪e

D4∪e

D4(0,0, a; 1,0,0,0,0; 1,0,0,0,0)

G2∪e

A2∪e

E6(0,0, a; 1,0,0; 1,0,0,0,0,0,0)

G2∪e

E6(0,0, a; 0,0,√3; 1,0,0,0,0,0,0).

G2∪e

G2{x1↔(0,0, a1; 0,0,√3; 0,0,√3; 0,0,√3; 0,0,√3)

x2↔(0,0, a2; 1,0,0; 1,0,0; 1,0,0; 1,0,0)}

Table 4.2.11: Strings for dimension n= 11

(i) Assume that σ∞=e

G2∪e

A8and that F=P2is as depicted below.

For a1, a2, a3∈ {√3,3}, write

x1↔(0,0, a1; 1,0,1,0,0,0,0,0,0),

x2↔(0,0, a2; 0,0,0,1,0,1,0,0,0),

x3↔(0,0, a3; 0,0,0,0,0,0,1,0,1).

As we require {x1, x2, x3}to be an admissible set, we derive that a:= a1=

a2=a3. For both a=√3 and a= 3, it turns out that the resulting

polyhedron is not of ﬁnite volume.

Next, we search for vectors yisuch that the set {x1, x2, x3, y}is admissible.

2. Proof of the main theorem

For a=√3, we ﬁnd eight vectors yilisted in Table 4.2.12. In addition, we

give the Lorentzian products ⟨yi, yj⟩when {yi, yj}is an admissible pair in

the Table 4.2.13. However, we see that none of the admissible sets gives rise

to a ﬁnite-volume polyhedron.

Admissible vectors for a=√3⟨x1, yi⟩ ⟨x2, yi⟩ ⟨x2, yi⟩

y1↔(1,0,0; 0,0,0,0,√3,0,0,√3,0) −√3 0 0

y2↔(1,0,0; 0,0,0,√3,0,0,0,0,√3) 0 0 0

y3↔(1,0,0; 0,0,√3,0,0,0,√3,0,0) 0 0 0

y4↔(1,0,0; 0,√3,0,0,0,0,0,√3,0) 0 −√3 0

y5↔(1,0,0; 0,√3,0,0,√3,0,0,0,0) 0 0 −√3

y6↔(1,0,0; √3,0,0,0,0,√3,0,0,0) 0 0 0

y7↔(√3,0,0; 0,1,1,0,1,1,0,1,1) 0 0 0

y8↔(√3,0,0; 1,1,0,1,1,0,1,1,0) 0 0 0

Admissible vectors for a= 3 ⟨x1, y⟩ ⟨x2, y⟩ ⟨x2, y⟩

(1,0,0; 0,0,0,0,1,0,0,1,0) -1 0 0

(1,0,0; 0,0,0,1,0,0,0,0,1) 0 0 0

(1,0,0; 0,0,1,0,0,0,1,0,0) 0 0 0

(1,0,0; 0,1,0,0,0,0,0,1,0) 0 -1 0

(1,0,0; 0,1,0,0,1,0,0,0,0) 0 0 -1

(1,0,0; 1,0,0,0,0,1,0,0,0) 0 0 0

Table 4.2.12: Admissible sets {x1, x2, x3, yi}for σF=P2

y2y3y4y5y6y7y8

y1- 1 -1 -1 -1 -1 0 0

y2-1 -1 -1 -1 0 0

y3-1 -1 -1 0 0

y4- 1 -1 0 0

y5-1 0 0

y60 0

Table 4.2.13: Lorentzian products ⟨yi, yj⟩

For a= 3, we ﬁnd six vectors ylisted in Table 4.2.12. Again, none of the

admissible sets {x1, x2, x3, y}leads to a polyhedron of ﬁnite volume.

Furthermore, there is no admissible set of cardinality >4.

(ii) Assume that σ∞=e

G2∪e

G2and that F=P⋆.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

For a1, a2, a3∈ {√3,3}, write

x1↔(0,0, a1; 0,0,√3; 0,0,√3; 0,0,√3; 0,0,√3) ,

x2↔(0,0, a2; 1,0,0; 1,0,0; 1,0,0; 1,0,0) .

As {x1, x2}should be an admissible pair, this forces a2to be equal to √3a1,

that is, a1=√3 and a2= 3. We obtain ⟨x1, x2⟩= 0. However, these data

do not yield a ﬁnite-volume polyhedron.

Next, we ﬁnd the following vectors y1, . . . , y4such that the pairs {x1, yi}and

{x2, yi}are admissible.

y1↔(1,0,0; 0,0,3; 1,0,0; 1,0,0; 1,0,0),

y2↔(1,0,0; 1,0,0; 0,0,3; 1,0,0; 1,0,0),

y3↔(1,0,0; 1,0,0; 1,0,0; 0,0,3; 1,0,0),

y4↔(1,0,0; 1,0,0; 1,0,0; 1,0,0; 0,0,3) .

They all are orthogonal to both x1and x2, and they are pairwise admissible.

However, there is no admissible set giving rise to a ﬁnite-volume polyhedron.

Assume that 12 ≤n≤18

There is nothing to check in dimension n= 12 as there is neither an ADE-

polyhedron nor an ADEG-polyhedron in H10 serving as a G2-face F.

For dimension n= 13, there are two possibilities for the face F. Either F

is an ADE-pyramid or an ADEG-pyramid. The strings are listed in Table

4.2.14. If σ∞=e

G2∪e

A2∪e

E8, we ﬁnd no admissible pair, and in the case

σ∞=e

G2∪e

E8, the admissible pairs are listed in Appendix B, Table

2.0.26. None of them yields a ﬁnite-volume polyhedron.

For dimension n= 14, Fhas to be the (unique) ADE-pyramid in H12. The

corresponding string is given in Table 4.2.14, and we ﬁnd no admissible pairs.

Hence, there is no ADEG-polyhedron.

For dimension n= 15, Fhas to be the (unique) ADE-pyramid in H13. The

corresponding string is given in Table 4.2.14, and again, we ﬁnd no admissible

pairs and therefore no ADEG-polyhedron.

For dimensions n= 16,17 and 18, there is neither an ADE-polyhedron nor

an ADEG-polyhedron in Hn−2serving as a G2-face F.

This ﬁnishes the proof for n≤18.

3. Further results and comments

n σ∞String for x

13 e

G2∪e

A2∪e

E8(0,0, a; 1,0,0; 0,0,0,0,0,0,0,0,1)

G2∪e

E8(0,0, a; 0,0,√3; 0,0,0,0,0,0,0,0,1)

14 e

G2∪e

A3∪e

E8(0,0, a; 1,0,0,0; 0,0,0,0,0,0,0,0,1)

15 e

G2∪e

D4∪e

E8(0,0, a; 1,0,0,0,0; 0,0,0,0,0,0,0,0,1)

Table 4.2.14: Strings for dimensions 12 ≤n≤18

Assume that n ≥19

Let n= 19. The single possibility for the G2-face F⊂H17 is given by the

ADE-pyramid depicted below.

We have σ∞=e

G2∪e

E8∪e

E8, and we write

x↔(0,0, a; 0,0,0,0,0,0,0,0,1; 0,0,0,0,0,0,0,0,1) , a ∈ {√3,3}.

For a= 3, there is a unique admissible pair which, however, does not yield a

ﬁnite-volume polyhedron; see Table 2.0.27 in Appendix B.

For a=√3, we ﬁnd 26 vectors ysuch that {x, y}is an admissible pair. They

are all listed in Table 2.0.27. Apart from a few exceptions, each admissible

set gives rise to a spherical subdiagram of rank >19. For the few exceptions

left, we do not ﬁnd any ﬁnite-volume polyhedron.

Finally, when n= 20, the inductive procedure stops, since there are no

ADEG-polyhedron in dimensions 18 and 19.

This ﬁnishes the proof of Theorem 4.0.2.

3 Further results and comments

3.1 Properties of the polyhedron P⋆

The ADEG-polyhedron P⋆⊂H9depicted in Figure 4.0.2 and its associated

Coxeter group Γ⋆= Γ(P⋆) have some interesting properties which we shall

develop in what follows.

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

Let us start with a rough combinatorial information about P⋆provided by

the software CoxIter [30, 31]. The f-vector of P⋆is given by

f= (134,671,1480,1909,1606,917,356,91,14) ,

where among the 134 vertices of P⋆, six of them are ideal; see Figure 4.3.3.

Figure 4.3.3: CoxIter’s output for P⋆

As a supplementary information, the growth rate τ⋆≈3.029708 of Γ⋆is a

Perron number whose minimal polynomial has degree 51 and integral coef-

ﬁcients in the interval [−217,283]. We will come back to growth rates and

Perron numbers in Part III.

As for the volume of P⋆, we can derive that

vol(P⋆)∈Q·ζ(5) ,(4.15)

where ζ(5) denotes Riemann’s zeta function evaluated at 5. In fact, the non-

cocompact group Γ⋆is arithmetic (over Q) by the criterion stated in Theorem

1.3.20, and Γ⋆acts on hyperbolic space of odd dimension ≥5. The property

(4.15) is now an immediate consequence of Emery’s Proposition 2.1 [19], in

view of the quadractic form q⋆associated to P⋆; see the proof of Proposition

4.3.1 below.

More precisely, we shall see that

vol(P⋆) = q·ζ(5)

22,295,347,200 with q∈Q>1.(4.16)

To this end, consider the Coxeter simplex ∆9⊂H9given by the Coxeter

diagram depicted in Figure 4.17.

3. Further results and comments

Figure 4.3.4: The Coxeter simplex ∆9⊂H9

Alike Γ⋆, the Coxeter group Γ9= Γ(∆9) is arithmetic (over Q), and it is dis-

tinguished by the property that among all cusped hyperbolic 9-orbifolds, the

space H9/Γ9has minimal volume and is as such unique; see [35]. Moreover,

the volume of H9/Γ9is given by the volume of ∆9and has been computed

in [40] according to

vol(∆9) = ζ(5)

22,295,347,200 .(4.17)

As a consequence, the identity (4.16) is veriﬁed.

Recall from Chapter 1 the concept of (widely) commensurable groups of hy-

perbolic isometries. Moreover, commensurability is an equivalence relation

which preserves properties such as arithmeticity, cocompactness and coﬁnite-

ness.

Our aim is to show that Γ⋆and Γ9are commensurable. We will prove even

more by looking also at the group Γ(P2) associated with the ADE-polyhedron

P2of Prokhorov in H9, depicted in Figure 3.2.8.

The group Γ(P2) is also arithmetic (over Q), and hence, we obtain

vol(P2) = q′·vol(∆9) with′∈Q>1.(4.18)

A natural question is whether all three groups Γ⋆, Γ(P2) and Γ9are com-

mensurable. In fact, this question can be answered positively as follows.

Proposition 4.3.1. The Coxeter groups Γ⋆,Γ(P2)and Γ9are pairwise com-

mensurable.

Proof. By Maclachlan’s work [57], there is a complete set of commensurability

invariants for arithmetic groups of hyperbolic isometries. His work has been

exploited by Guglielmetti, Jacquemet and Kellerhals [33] who established

these invariants in many cases and also for the group Γ9. Hence, it is suﬃcient

to show that the sets of invariants for Γ⋆and Γ(P2) coincide with the one of

Γ9.

Since we are in dimension 9, which is odd, and since all three groups un-

der consideration are deﬁned over Q, the set of commensurability invariants

according to Maclachlan can be described brieﬂy as follows. First, one es-

tablishes the Gram matrix Gand the diﬀerent cycle coeﬃcients for each

Chapter 4. Coxeter polyhedra with dihedral angles π

2,π

3and π

Coxeter polyhedron in terms of its normal vectors and passes then to the

Vinberg form qassociated to G; see Deﬁnition 1.3.19 and Appendix C.

Then, one computes the signed determinant δ=−det(q) of qin Q/Q2. The

next step is to determine the Hasse invariant s(q) and the Witt invariant c(q)

of q. More precisely, by expressing q=< a1, a2, . . . , a10 >in diagonal form,

one has

s(q) = O

1≤i<j≤10

(ai, aj)

in terms of the diﬀerent quaternion algebras (ai, aj). In our case, s(q) and

c(q) coincide. Choose a quaternion algebra Brepresenting c(q), and ﬁnally

establish the ramiﬁcation set Ramq(B) resp. Ramq(B⊗QQ(√δ)) according

to whether δis a square in Qor not.

The complete set of commensurability invariants is then given by

{Q, δ, Ramq(B)}if δis a square in Q;

{Q, δ, Ramq(B⊗QQ(√δ))}if δis not a square in Q.

⋄For the group Γ9, the invariant set is known and equals to {Q,1,∅}.

⋄For the group Γ⋆, the Vinberg form is given by

q⋆=<1,1,2,3,3,6,6,10,10,−2>;

see Appendix C for its computation. Therefore, the signed determinant of q⋆

equals 1 in Q/Q2.

By the properties of quaternion algebras, the Hasse invariant s(q⋆) given by

(2,−2) ·(3,3) ·(6,6) ·(10,10) can be identiﬁed with (5,−1).

We derive that the ramiﬁcation set of (5,−1) is empty, and that the complete

set of invariants for Γ⋆is given by {Q,1,∅}.

⋄For the group Γ(P2) we proceed in a similar way. The Vinberg form,

established in Appendix C, is given by

q2=<1,1,3,6,7,10,10,15,21,−10 > ,

which yields 1 for the signed determinant of q2. Again, we obtain c(q2) =

(5,−1), whose ramiﬁcation set is empty. Therefore, the set of invariants for

Γ(P2) is given by {Q,1,∅} as well.

We conclude that all three groups Γ9, Γ⋆and ΓP2are pairwise commensurable.

3. Further results and comments

Remark 4.3.2. By [33, 41] and Proposition 4.3.1, it follows that the com-

mensurability class of Γ⋆= Γ(P⋆) contains all Coxeter simplex groups and

all Coxeter pyramid groups in IsomH9.

3.2 An angular obstruction

In this section, we present an angular obstruction to the existence of hyper-

bolic Coxeter polyhedra with mutually intersecting facets.

The following result provides a universal bound from below for dihedral an-

gles of Coxeter polyhedra with mutually intersecting facets in dimensions

beyond 6.

Proposition 4.3.3. Let n≥7, and let P⊂Hnbe a Coxeter polyhedron with

mutually intersecting facets. Then, any dihedral angle of Pis of the form π

with m≤6.

Proof. Let n≥7, and assume that all facets of Coxeter polyhedron P⊂Hn

are mutually intersecting. Let Σ be the Coxeter diagram of P, and denote

by σm= [m] a subdiagram of type G(m)

2in Σ. Suppose that Phas a dihedral

angle π

mfor m≥7.

As in the proof of Lemma 4.2.1 and by means of Theorem 1.3.9, we deduce

that the G(m)

2-face F=F(σm) of Pis an (n−2)-Coxeter polyhedron with

mutually intersecting facets.

As n−2≥5, the face Fis a non-compact Coxeter polyhedron; see Theorem

3.2.2 and Theorem 3.2.3. Therefore, its Coxeter diagram σFcontains an aﬃne

subdiagram σ∞

Fof rank n−3, which is a component of an aﬃne subdiagram

of rank n−1 in Σ. As Σ \σ∞

Fcontains σm, which cannot be extended to

yield an aﬃne component, we get a contradiction.

Final comments. The method we developed in order to classify all ADEG-

polyhedra is general and can be applied to Coxeter polyhedra with mutually

intersecting facets whose dihedral angles are prescribed.

By Proposition 4.3.3, for dimensions ≥7, all dihedral angles of such poly-

hedra are uniformly bounded from below by π

6. Therefore, it is realistic to

obtain a complete classiﬁcation of all Coxeter polyhedra with mutually inter-

secting facets and for all dimensions in ﬁnite time. Observe, however, that

the presence of a dihedral angle π

4makes the classiﬁcation task much more

voluminous.

My work is a game, a very serious game.

M.C. Escher

Part III:

On the growth rates of

hyperbolic Coxeter groups

CHAPTER 5

The growth rates of Coxeter groups

In this chapter, we provide the necessary background for the study of growth

rates of Coxeter groups. We start with basic deﬁnitions about the growth

series and growth rates of Coxeter groups. Then we present the tools for

comparing growth rates of Coxeter groups that will be fundamental for the

next chapter about growth minimality. Main references for this chapter are

[17, 37, 60, 76].

1 Growth series and growth rates

Let Γ = (W, S) be an abstract Coxeter group of rank Ngenerated by S,

presented as in (1.9).

Denote by lSthe length function of Γ with respect to S, that is, for w∈W,

lS(w) = min{k∈Z>0| ∃s1, . . . , sk∈S , w =s1···sk}, lS(1) = 0 .(5.1)

Deﬁne ak∈Nto be the number of elements w∈Wwith S-length k, that is,

ak={w∈W|lS(w) = k} ∈ Z[t].(5.2)

Deﬁnition 5.1.1. The growth series fS(t) of Γ = (W, S) is the power series

deﬁned by

fS(t) = 1 + X

k≥1

aktk.(5.3)

Chapter 5. The growth rates of Coxeter groups

Notice that if Γ = (W, S) is ﬁnite, then fS(t) is a polynomial. More speciﬁ-

cally, by a classical result of Solomon [74], the growth polynomial fS(t) can

be expressed by means of its exponents {m1, m2, . . . , mn}according to the

formula

fS(t) =

i=1

[mi+ 1] ,(5.4)

where [m] = 1 + t+. . . +tm−1and [m1, . . . , mr] := [m1]···[mr] . The

exponents of all irreducible spherical Coxeter groups are listed in Table 5.1.1.

Group Exponents Group Exponents

An1,2,··· , n E61,4,5,7,8,11

Bn1,3,··· ,2n−1E71,5,7,9,11,13,17

Dn1,3,··· ,2n−3, n −1E81,7,11,13,17,19,23,29

H31,5,9G(m)

21, n −1

H41,11,19,29 F41,5,7,11

Table 5.1.1: Exponents of the irreducible spherical Coxeter groups

The following well-known formula due to Steinberg [75] allows one to

express the growth series fS(t) of an arbitrary Coxeter group (W, S) in terms

of its ﬁnite parabolic subgroups.

Theorem 5.1.2 (Steinberg’s formula).

fS(t−1)=X

WT<W

|WT|<∞

(−1)|T|

fT(t),(5.5)

where W∅={1}.

As a consequence, in its disk of convergence, the growth series fS(t) is a

rational function, which can be expressed as the quotient of two coprime

monic polynomials p(t), q(t)∈Z[t] of the same degree.

Deﬁnition 5.1.3. The growth rate τΓ=τW:= τ(W,S)is deﬁned as the inverse

of the radius of convergence of the growth series fS(t) of (W, S).

1. Growth series and growth rates

By Hadamard’s formula, we have

τΓ= lim sup

k→∞

ak1/k .(5.6)

Furthermore, the growth rate τΓcan be identiﬁed with the inverse of the

smallest positive real pole of fS(t). Hence, τΓis an algebraic integer.

For a reducible group, the growth series fS(t) satisﬁes the following product

formula.

Proposition 5.1.4. The growth series of a reducible Coxeter group (W, S)

with factor groups (W1, S1),(W2, S2)such that S= (S1×{1W2})∪({1W1}×S2)

satisﬁes

fS(t) = fS1(t)·fS2(t).

In particular, τW= max

i=1,2τWi.

Let Γ ⊂IsomXnbe a (coﬁnite) geometric Coxeter group, as usually with

its canonical system Sof generating reﬂections; see (1.9). If Xn=Sn, then

τΓ= 0, while for Xn=En,τΓ= 1. In both cases, Γ is of polynomial growth.

For the case Xn=Hn, results of Milnor [63] and de la Harpe [17] imply that

Γ has exponential growth, that is, τΓ>1. More precisely, it was proven by

Terragni [76, 77] that for any hyperbolic Coxeter group Γ,

τΓ≥τΓ9≈1.138078 ,(5.7)

where Γ9is the Coxeter 9-simplex group depicted in Figure 5.1.1; see also

Chapter 4.

Figure 5.1.1: The Coxeter simplex group Γ9⊂IsomH9

Example 5.1.5. Let Γ = [7,3] be the cocompact hyperbolic triangle group

with Coxeter diagram

Recall that the group Γ has smallest co-area among all discrete groups in

IsomH2; see Chapter 3.

As easily read from the Coxeter diagram above, Γ contains six ﬁnite parabolic

subgroups. Namely, it contains the three dihedral subgroups A1×A1, A2and

Chapter 5. The growth rates of Coxeter groups

G(7)

2, each with multiplicity 1, and the subgroup A1with multiplicity 3. By

means of Steinberg’s formula (5.5) and Table 5.1.1, we derive

fS(t−1)= 1 −3

[2] +1

[2,2] +1

[2,3] +1

[2,7] .

It follows that

fS(t) = C(t)

L(t)=C(t)

1 + t−t3−t4−t5−t6−t7+t9+t10 (5.8)

where C(t) is a certain product of cyclotomic polynomials, and where L(t)

is Lehmer’s polynomial. It is the minimal polynomial of the Lehmer number

αL≈1.17628, which is the smallest known Salem number; see Section 2. As

a result of (5.8), we derive τ[7,3] =αL.

For dimensions n= 2 and 3, there are closed formulas for the growth series

of coﬁnite hyperbolic Coxeter groups as follows.

Let us ﬁrst discuss the 2-dimensional cocompact case.

Theorem 5.1.6 (Floyd, Plotnick [28]).Let P⊂H2be a compact Coxeter

N-gon with interior angles π

k1,..., π

kN. Let Γbe the planar hyperbolic Coxeter

group associated with P. Then, the growth series of Γis given by

fS(t) = [2, k1, . . . , kN]

(t−N+ 1)[k1, . . . , kN] + PN

i=1[k1, . . . , ki−1][ki+1, . . . , kN].

As for a formula in dimension 3, let us recall that compact Coxeter polyhedra

are simple. Therefore, given a compact Coxeter polyhedron Pin H3, each

vertex vof Pis the intersection of precisely three hyperplanes, giving rise

to the stabilizer Γvwhich is a spherical Coxeter subgroup of rank 3 in Γ.

Denote by m1= 1, m2and m3the exponents of Γvaccording to Table 5.1.1.

Theorem 5.1.7 (Parry [67]).Let P⊂H3be a compact Coxeter polyhe-

dron with associated Coxeter group Γ. Then, the growth series of Γcan be

expressed as

fS(t) = t−1

t+ 1 −1

2t(t−1) X

v∈P

(tm1−1)(tm2−1)(tm3−1)

(tm1+1 −1)(tm2+1 −1)(tm3+1 −1) .

1. Growth series and growth rates

In the non-cocompact coﬁnite case, similar results for dimensions n= 2 and

3 are available and due to Floyd [29] and Kellerhals [46].

For a non-compact polyhedron P⊂Hn, denote by f0=fo

0+f∞

0the number

of vertices of P, where fo

0is the number of ordinary vertices, and f∞

0is the

number of ideal vertices of P. When n≥3, ideal vertices may be non-simple.

Again, the stabilizer of an ordinary vertex vof Pis a ﬁnite subgroup Γvof

Γ, generated by the reﬂections in the hyperplanes passing through v.

For the 2-dimensional case, there is the following result.

Theorem 5.1.8 (Floyd [29]).Let P⊂H2be a non-compact Coxeter polygon

of ﬁnite volume with f0=fo

0+f∞

0vertices. Let Γbe the hyperbolic Coxeter

group associated with P. Then, the growth series of Γsatisﬁes

fS(t)= 1 −t

[2](f∞

0+X

vordinary

[kv−1]

[kv]),

where 2kvis the order of the dihedral stabilizer group Γvof the ordinary vertex

vof P.

As for the 3-dimensional case, some preparation is necessary.

Let P⊂H3be a non-compact Coxeter polyhedron with associated Coxeter

group Γ and Coxeter diagram Σ. If Phas an ordinary vertex v, its stabilizer

Γvin Γ is a spherical Coxeter group with exponents m1= 1, m2and m3

as above. As for the ideal vertices of P, they can be either simple or non-

simple. Suppose that Phas a non-simple vertex v∞. Then, its subdiagram

σ∞in Σ is of type f

A1×f

A1. Now, the polyhedron Pcan be obtained by a

deformation process from a sequence of Coxeter polyhedra Pk⊂H3which

all have the same combinatorial structure as P, apart from a ﬁnite edge with

vertices both of type A1×A1×G(k)

2that collapses to yield the vertex v∞

when k→ ∞. As for the corresponding growth functions f(t) of Γ and fk(t)

of Γ(Pk), a result of Kolpakov [50] shows that

f(t)=1

fk(t)+tk

tk−1t−1

t+ 12

for all k .

As a consequence, it is primarly of interest to have a closed formula for the

growth series of a non-compact hyperbolic Coxeter polyhedron with simple

vertices, only.

Theorem 5.1.9 (Kellerhals [46]).Let P⊂H3be a non-compact simple

Coxeter polyhedron of ﬁnite volume with associated Coxeter group Γ. Then,

Chapter 5. The growth rates of Coxeter groups

the growth series of Γsatisﬁes

fS(t)=1−t

1 + t1−t

2X

vordinary

[m2, m3]

[m2+ 1, m3+ 1] +X

wideal

[nw−1] + tnw−2

[nw]

where

nw= max (nw

1, nw

2, nw

3) = 









3if Γw=e

4if Γw=e

6if Γw=e

is deﬁned in terms of the dihedral angles π/nw

iat the three edges giving rise

to the simple ideal vertex w, and where Γwis the stabilizer of w.

Let us point out that one does not dispose up to now of similar closed formulas

for growth series of Coxeter groups Γ ⊂IsomHnfor n≥4, except for the case

related to right-angled compact Coxeter polyhedra; see the work of Kellerhals

and Perren [45].

2 About the arithmetic nature of growth rates

In this section we discuss the arithmetic nature of the growth rate of a (coﬁ-

nite) hyperbolic Coxeter group Γ = (W, S). Recall that the growth rate τΓis

an algebraic integer which can be identiﬁed with the inverse of the smallest

positive real pole of the growth series fS(t).

Deﬁnition 5.2.1. The growth series fS(t) is said to be reciprocal if it satisﬁes

fS(t−1) = fS(t). It is called anti-reciprocal if fS(t−1) = −fS(t).

In the cocompact case, it is known by works of Charney and Davis [14] that

the growth series is reciprocal in even dimensions, and anti-reciprocal in odd

dimensions. However, this property does not hold in the non-cocompact case

anymore.

Let us now consider growth rates of hyperbolic Coxeter groups. In low

dimensions, they appear to be Salem numbers, Pisot numbers (or Pisot-

Vijayaraghavan numbers), or Perron numbers. These algebraic integers are

deﬁned as follows.

Deﬁnition 5.2.2. An algebraic integer τ > 1 is a Salem number if it is

either a quadratic unit, or its inverse τ−1is a Galois conjugate of τand the

other Galois conjugates lie on the unit circle.

2. About the arithmetic nature of growth rates

Deﬁnition 5.2.3. An algebraic integer τ > 1 is a Pisot number if τis either

an integer, or if all of its other Galois conjugates are contained in the unit

open disk.

Deﬁnition 5.2.4. An algebraic integer τ > 1 is a Perron number if τis

either an integer, or if all of its other Galois conjugates are strictly less than

τin absolute value.

In particular, Salem numbers and Pisot numbers are Perron numbers.

Let us turn back to hyperbolic Coxeter groups.

In [67], Parry proved that the growth rate of any cocompact hyperbolic

Coxeter group in IsomHnis a Salem number when n= 2 or 3.

The smallest known Salem number is the Lehmer number αLwhich is also the

growth rate of the Coxeter triangle group [7,3]; see Example 5.1.5. However,

by a result of Kellerhals and Liechti [44], not every Salem number appears

as the growth rate of a cocompact hyperbolic Coxeter group.

In the non-cocompact case, Floyd [29] proved that the growth rates of coﬁ-

nite hyperbolic Coxeter groups in IsomH2are Pisot numbers. However, the

analogous result does not hold anymore in IsomH3; see [46] for instance.

Nevertheless, the growth rate of any coﬁnite hyperbolic Coxeter group in

IsomH3is a Perron number. This result was proven in the special case of

ideal Coxeter polyhedra by Komori and Yukita [52], as well as by Nonaka

and Kellerhals [66]. Soon later, it was proven by Yukita [86] in full generality.

Remark 5.2.5. Related is our joint work with Yukita [9], wherein we ex-

tended the results of Parry [67] and Floyd [29] to abstract Coxeter groups

having 2-dimensional Davis complexes. Namely, we proved that if the Euler

characteristic χof the nerve of a Coxeter group is zero, then its growth rate

is a Salem number. If χis positive, then its growth rate is a Pisot num-

ber. For negative Euler characteristic, we provided inﬁnitely many families

of Coxeter groups of index of inertia >1 whose growth rates are Perron

numbers. These families contain so-called ∞-spanned Coxeter groups whose

growth rates have already been identiﬁed with Perron numbers by Kolpakov

and Talambutsa [51].

However, in higher dimensions, only some partial results are available but

several questions arise.

The growth rates of cocompact Coxeter groups in IsomH4of rank at most 6

are Perron numbers, as shown by Kellerhals and Perren [45]. The following

problem extends their conjecture in the cocompact case to the coﬁnite setting

in a natural way.

Chapter 5. The growth rates of Coxeter groups

Conjecture. Let Γ⊂IsomHnbe a coﬁnite Coxeter group. Then the growth

rate τΓwith respect to the natural set of generating reﬂections of Γis a

Perron number.

A proof of this Conjecture seems to be hard. One reason is the lack of explicit

formulas for the growth series for n > 3. Other reasons are the number-

theoretical diﬃculties concerning the irreducibility of integer polynomials.

Nevertheless, for any known Coxeter group Γ ⊂IsomHn, the software Math-

ematica [85] or the program CoxIter of Guglielmetti [30, 31] allows one to

test whether the growth rate τΓis a Perron number.

Example 5.2.6. Consider the Coxeter group Γ⋆= Γ(P⋆) in IsomH9. By the

software CoxIter, the minimal polynomial of the growth rate τ⋆≈3.029708

of Γ⋆has degree 51. The program shows that τ⋆is neither a Salem number

nor a Pisot number, but it is a Perron number; see Figure 4.3.3.

3 Comparing growth rates

In this section, our goal is to compare growth series and growth rates of

abstract Coxeter groups. This will be essential when studying minimality

aspects of growth rates of coﬁnite hyperbolic Coxeter groups.

3.1 Partial order and growth monotonicity

In [60], McMullen introduced the following partial order on the set of Coxeter

groups.

Let (W, S) and (W′, S′) be two Coxeter groups. Deﬁne (W, S)≤(W′, S′) if

there exists an injective map ι:S→S′such that

mst ≤m′

ι(s)ι(t)for all s, t ∈S.

As mst ∈ {2,3,...,∞} for all distinct s, t ∈S, this partial order satisﬁes the

descending chain condition. Therefore, any strictly decreasing sequence of

Coxeter groups is ﬁnite.

In addition, one deﬁnes (W, S)<(W′, S′) if (W, S)≤(W′, S′) and ιcannot

be extended to an isomorphism. In other words, one has (W, S)<(W′, S′)

if |S|<|S′|, or if mst < m′

ι(s),ι(t)for some s, t ∈S,s=t.

The following result, established by Terragni [76, 77], is of major importance

for the subsequent study of minimal growth rates.

3. Comparing growth rates

Theorem 5.3.1. If (W, S)≤(W′, S′), then τ(W,S)≤τ(W′,S′).

In particular, the growth rate of any parabolic subgroup (WT, T ) of (W, S)

satisﬁes τ(WT,T )≤τ(W,S).

Remark 5.3.2. Terragni’s growth monotonicity result stated in Theorem

5.3.1 is really convenient for the comparison of growth rates. However, this

monotonicity property is not proven in its strict sense. Indeed, Terragni’s

proof is based on Hadamard’s limit formula (5.6) for the coeﬃcients in the

respective growth series which does not allow one to deduce the strict mono-

tone behaviour as we observed in practice. Motivated by this, we formulate

the following conjecture.

Conjecture. Let (W, S),(W′, S′)be two non-spherical, non-aﬃne irreducible

Coxeter groups such that (W, S)<(W′, S′). Then,

τ(W,S)< τ(W′,S′).(5.9)

Remark 5.3.3. The conjectured strict monotonicity property (5.9) for the

growth rate of a Coxeter group Γ ⊂IsomHncomes from the comparison with

the strict monotonicity of the volume, with respect to the dihedral angles, of

the Coxeter polyhedron associated with Γ; see Remark 1.2.5.

As a preparation for the subsequent investigations, let us introduce the notion

of extension of a Coxeter diagram.

Deﬁnition 5.3.4. Let Σ be a Coxeter diagram. An extension of Σ is a

Coxeter diagram Σ′obtained by adding one node linked with a (labelled)

edge eto Σ. The extension Σ′is said to be simple if the edge ehas no label.

Let Wbe a Coxeter group with Coxeter diagram Σ. By deﬁnition, any

extension Σ′of Σ encodes a Coxeter group W′such that W < W ′. Hence,

by Theorem 5.3.1, we get τW≤τW′.

Furthermore, if Σ′

0is a simple extension of Σ by an edge e, the extension

Σ′of Σ obtained by adding a label on the edge eof Σ′

0yields the following

inequality.

τΣ≤τΣ′

0≤τΣ′,

where, by abuse of notation, the growth rate of Σ equals the growth rate of

its Coxeter group W.

Next, we reproduce the statements concerning the simple extensions of aﬃne

Coxeter diagrams of small ranks stated in [8].

Chapter 5. The growth rates of Coxeter groups

Example 5.3.5. Consider a connected aﬃne Coxeter diagram Σ of rank n;

see Table 1.3.2. Then, one veriﬁes that any extension Σ′of Σ corresponds to

a non-cocompact Coxeter simplex group in IsomHn+1. However, notice that

the resulting group might not be coﬁnite.

Nevertheless, we observe that for rank n≤3, the resulting Coxeter simplex

groups in IsomHn+1 are always coﬁnite; see Theorem 3.1.1 and Table 3.2.1.

If Σ = [∞] is of rank 1, there is a unique simple extension, up to symmetry,

and it corresponds to the Coxeter triangle group [∞,3]. For ranks 2 and 3,

all resulting simple extensions are depicted in Figure 5.3.2 and Figure 5.3.3.

4 4 4 4

6 6 6

Figure 5.3.2: The simple extensions of e

A2,e

C2and e

4 4

4 4 4 4

Figure 5.3.3: The simple extensions of e

A3,e

B3and e

In the case where Σ is aﬃne of rank n≥4, some simple extensions of Σ

yield hyperbolic groups of inﬁnite covolume. Note that for ranks ≥9, any

extension gives rise to a group of inﬁnite covolume.

4 4 4 4 4

Figure 5.3.4: The Coxeter groups ∆1,...,∆4of inﬁnite covolume

For rank 4, there are ﬁfteen simple extensions of connected aﬃne Coxeter

diagrams. There are exactly eleven simple extensions each giving rise to a

coﬁnite Coxeter simplex group in IsomH5, and there are four more extensions

corresponding to simplex groups ∆i, 1 ≤i≤4, of inﬁnite covolume depicted

in Figure 5.3.4.

3. Comparing growth rates

3.2 A useful lemma

Let Γ = (W, S) and Γ′= (W′, S′) be two abstract Coxeter groups. By

means of Steinberg’s formula (5.5), one can express their growth series in an

eﬀective way allowing one to derive the following lemma which provides a

tool to compare the growth rates of Γ and Γ′.

Lemma 5.3.6. Assume that for all t > 0, one has

fS(t−1)−1

fS′(t−1)>0.(5.10)

Then, τΓ< τΓ′.

Proof. The inequality (5.10) implies that for all x=t−1∈(0,1), the smallest

zero of the function g(x) := 1

fS(t−1)is strictly bigger than the smallest zero of

g′(x) := 1

fS′(t−1). As these two zeros correspond to the radii of convergence of

the growth series fS(t) and fS′(t), when passing from xto t, we immediately

deduce that their growth rates satisfy τΓ< τΓ′.

We now reproduce in Examples 5.3.7 and 5.3.8 two applications of Lemma

5.3.6 taken from the articles [7, 8].

Example 5.3.7. Consider the Coxeter groups (W1, S1),(W2, S2) and (W3, S3)

given by simple extensions of [∞,3] as follows.

W1∞W2∞W3∞

By Steinberg’s formula (5.5), we compute

fS1(t−1)= 1 −4

[2] +3

[2,2] +2

[2,3] −1

[2,2,3] −1

[2,3,4] ,

fS2(t−1)= 1 −4

[2] +3

[2,2] +2

[2,3] −2

[2,2,3] ,

fS3(t−1)= 1 −4

[2] +3

[2,2] +2

[2,3] −1

[2,2,2] −1

[2,3,4] .

We get positive diﬀerence functions for all t > 0, according to

fS1(t−1)−1

fS2(t−1)=1

[2,2,3] −1

[2,3,4] =t2+t3

[2,2,3,4] ,

Chapter 5. The growth rates of Coxeter groups

fS1(t−1)−1

fS3(t−1)=1

[2,2,2] −1

[2,2,3] =t2

[2,2,2,3] .

By Lemma 5.3.6, we deduce that τW1< τW2and τW1< τW3.

Example 5.3.8. Consider the Coxeter groups (W1, S1),(W2, S2) and (W3, S3)

given by (∞-labelled) extensions of [∞,3].

W1∞ ∞ W2∞ ∞ W3∞ ∞

By means of Steinberg’s formula (5.5), we have

fS1(t−1)= 1 −4

[2] +3

[2,2] +1

[2,3] ,

fS2(t−1)=1

fS1(t−1)−1

[2,2,3] ,

fS3(t−1)=1

fS1(t−1)−1

[2,2,2] .

As above, Lemma 5.3.6 implies that τW1< τW2and τW1< τW3.

CHAPTER 6

Minimal growth rates for

hyperbolic Coxeter groups

Let Γ ⊂IsomHnbe a coﬁnite Coxeter group. Let τΓbe its growth rate with

respect to the canonical system Sof generators for Γ; see (1.9) and Chapter

5. Then, τΓis always bounded from below by the growth rate τΓ9≈1.138078

of the simplex group Γ9; see Figure 5.1.1 and (5.7).

In this chapter, we are interested in providing a sharp lower bound, for ﬁxed

dimension n, for the growth rate of a hyperbolic Coxeter group in IsomHn.

First, we discuss the low dimensional cases where we dispose of results due to

Hironaka [36] and Kellerhals–Kolpakov [43] in the cocompact case, and due

to Floyd [29] and Kellerhals [46] in the non-cocompact case. An important

observation, motivating our work in higher dimensions, is the following one.

In all known cases, the group achieving minimal growth rate is a Coxeter

simplex group, and it is distinguished by the fact of being closely related

to the fundamental group of minimal covolume when considering (compact,

respectively cusped) hyperbolic orbifolds.

The next section is devoted to the study of minimal growth rates in higher

dimensions, where we reproduce our results in the joint work with Kellerhals

[7] in the cocompact case, and in our work [8] in the non-cocompact case.

Chapter 6. Minimal growth rates for hyperbolic Coxeter groups

1 In low dimensions

It is well known by Siegel’s work [73] that the group of minimal co-area

among all discrete groups in IsomH2is the planar Coxeter group Γc

2= [7,3].

In the non-cocompact case, the smallest co-area is achieved by the Coxeter

triangle group Γ2= [∞,3] intimately related to the modular group SL2(Z).

In dimension 3, it was proven by Martin and his collaborators that the Z2-

extension of the Coxeter group Γc

3= [3,5,3] gives rise to the smallest volume

compact hyperbolic orbifold; see [59] for example. For cusped 3-orbifolds,

Meyerhoﬀ [62] showed that the smallest volume is achieved by the orbifold

H3/Γ3, where Γ3is the Coxeter group with Coxeter symbol [6,3,3].

These Coxeter simplex groups are also distinguished by the fact of having

minimal growth rates among all Coxeter groups acting on hyperbolic space.

In the cocompact case, this is due to Hironaka [36] for dimension 2, and to

Kellerhals and Kolpakov [43] for dimension 3. More precisely, their results

are as follows.

Theorem 6.1.1. Let n= 2 or 3. Among all cocompact Coxeter groups in

IsomHn, the Coxeter simplex group Γc

ndepicted in Figure 6.1.1 has minimal

growth rate, and as such it is unique.

Γc

7Γc

Figure 6.1.1: The cocompact Coxeter simplex groups Γc

2and Γc

In the non-cocompact case, as similar result is due to Floyd [29] in dimension

2, and to Kellerhals [46] in dimension 3.

Theorem 6.1.2. Let n= 2 or 3. Among all non-cocompact coﬁnite Coxeter

groups in IsomHn, the Coxeter simplex group Γndepicted in Figure 6.1.2 has

minimal growth rate, and as such it is unique.

Γ2∞Γ3

Figure 6.1.2: The non-cocompact Coxeter simplex groups Γ2and Γ3

The results stated in the previous theorems rely upon explicit formulas for

the growth series in dimension 2 and 3, as described in Theorems 5.1.6, 5.1.7,

5.1.8 and 5.1.9. Since in higher dimensions, no such formulas exist, up to

now, one needs to develop another, new, strategy to identify minimal growth

rate.

2. In higher dimensions

2 In higher dimensions

Motivated by Theorems 6.1.1 and 6.1.2, and in view of Remarks 1.2.5 and

(5.9), it is guessed that small covolume and small growth rate are intimately

related.

In this context, let us mention that in general compact hyperbolic n-orbifolds

of minimal volume are not known for dimensions n≥4. However, by restrict-

ing to arithmetically deﬁned (orientable) hyperbolic n-orbifolds, minimizers

for the volume have been detected, for instance, for n= 4 and 5, by ex-

ploiting Prasad’s formula. These results are due to Belolipetsky [3, 4] and to

Emery and Kellerhals [18]; see also [47, 48].

More precisely, among all arithmetic compact hyperbolic 4-orbifolds, the

fundamental group of minimal covolume is given by the Coxeter simplex

group Γc

4= [5,3,3,3]. In dimension 5, the minimizing group is the Coxeter

prism group Γc

5= [5,3,3,3,3,∞]. Here, the component ∞in the Coxeter

symbol of Γc

5symbolizes the presence of a pair of ultraparallel facets in the

underlying Coxeter prism.

In the non-compact case, cusped hyperbolic orbifolds of minimal volume are

known up to dimension 9. This result was established by Hild and Kellerhals

[34] in dimension 4, and by Hild [35] for dimensions up to 9. They showed

that the cusped n-orbifolds of minimal volume are related to the Coxeter

simplex groups Γn⊂IsomHn(up to a Z2-extension for n= 7) given in

Figure 6.2.3.

Γ4

4Γ5

Γ6

4Γ7

Γ8Γ9

Figure 6.2.3: The non-cocompact Coxeter n-simplex groups Γn⊂IsomHn

Let us now announce our two main results, proved in [7, 8], concerning min-

Chapter 6. Minimal growth rates for hyperbolic Coxeter groups

imal growth rate for coﬁnite Coxeter groups in IsomHnfor n≥4, which

complement the volume minimality results described above.

In a joint work with Kellerhals [7], we established the following result in the

cocompact case.

Theorem 6.2.1 (Bredon, Kellerhals [7]).Let n= 4 or 5. Among all cocom-

pact Coxeter groups in IsomHn, the Coxeter group Γc

ngiven in Figure 6.2.4

has minimal growth rate, and as such the group is unique.

Γc

5Γc

5∞

Figure 6.2.4: The cocompact Coxeter groups Γc

4and Γc

In the non-cocompact case, we were able to extend the tools developed for

the proof of Theorem 6.2.1 in a suitable way in order to derive the following

result.

Theorem 6.2.2 (Bredon [8]).Let 4≤n≤9. Among all non-cocompact

Coxeter groups of ﬁnite covolume in IsomHn, the Coxeter simplex group Γn

given in Figure 6.2.3 has minimal growth rate, and as such the group is

unique.

3 Proofs of the two main results

In this section, we describe the keys arguments for the proofs of Theorem

6.2.1 and Theorem 6.2.2. In Appendix D, the original articles [7] and [8] are

attached.

Let n≥4, and let Γ ⊂IsomHnbe a coﬁnite hyperbolic Coxeter group.

Denote by P⊂Hnthe associated Coxeter polyhedron and by Σ the Coxeter

diagram of Γ and P.

If Pis simple, which is always the case when Pis compact, we dispose of some

explicit results of Felikson and Tumarkin about the combinatorial structure

of P; see Section 2.2 in Chapter 3. The presence of a pair of disjoint facets

yields a subdiagram [∞] in Σ whose simple extensions will play an important

role.

3. Proofs of the two main results

If Pis non-simple, then Phas at least one non-simple ideal vertex which

comes with a disconnected aﬃne subdiagram of Σ. In this case, we consider

simple extensions of its components of small rank.

In both situations, these extension diagrams with their growth rates, when

combined with Terragni’s monotonicity result, allow us to complete the proofs

of Theorem 6.2.1 and Theorem 6.2.2.

3.1 The cocompact case

By Theorem 3.2.2 of Felikson and Tumarkin, there are only ﬁnitely many

compact Coxeter polyhedra P⊂Hn,n≥4, that do not contain any pair

of disjoint facets. In fact, they exist only for n= 4 and consist of the ﬁve

compact simplices and the seven Esselmann polyhedra; see Figure 3.2.4 and

Figure 3.2.5 for their diagrams.

It is known that the growth rate of Γc

4is minimal among all Coxeter simplices;

see [45], for example.

For the Esselmann groups E1, . . . , E7, we observe that they all contain a

cocompact hyperbolic triangle subgroup with Coxeter symbol [8,3],[10,3],

[5,4],[5,5] or given by a cyclic Coxeter diagram of order 3 with label set

{4,3,3}or {5,3,3}, respectively. It is easy to verify that the group [8,3] has

minimal growth rate among all of them, and that it satisﬁes

τΓc

4< τ[8,3] .(6.1)

By Theorem 5.3.1 of Terragni, and by inequality (6.1), we derive that

τΓc

4< τ[8,3] ≤τEi

for all i= 1,...,7.

Assume now that P⊂Hnhas at least one pair of ultraparallel facets, and

suppose that its Coxeter group Γ is not equal to Γc

4. It follows that the

Coxeter diagram Σ of Pnecessarily contains a subdiagram σof the form

∞

where p > 2 and q∈ {2,3,...,∞}. As [8,3] ≤[∞,3] ≤σ≤Γ by the partial

ordering of Coxeter groups, we derive by the same argument as above that

τΓc

4< τ[3,8] ≤τ[3,∞]≤τσ≤τΓ.

This proves the ﬁrst part of Theorem 6.2.1 about the group Γc

Chapter 6. Minimal growth rates for hyperbolic Coxeter groups

For the second part about n= 5, observe that P⊂H5has at least 7 facets,

and Pis either one of the two simplicial prisms K1, K2found by Kaplinskaya,

or Phas at least 8 facets; see Chapter 3. A direct computation shows that

τΓc

5< τKifor i= 1,2.

In the case where Phas at least 8 facets, Theorem 3.2.1 of Felikson and

Tumarkin implies that Phas at least two pairs of disjoint facets.

Hence, we deduce that Σ necessarily contains a subdiagram σof the form

∞

where p, q, r, s, t ∈ {2,3,...,∞}, and where at least one of them is greater

than 2 and at least one of them equals ∞.

For comparison, consider the diagram [∞,3,∞], which satisﬁes the inequality

τΓc

5< τ[∞,3,∞]. This can be seen by a direct computation.

By means of Theorem 5.3.1 and Example 5.3.8, we derive that

τΓc

5< τ[∞,3,∞]≤τσ≤τΓ,

and the proof of Theorem 6.2.1 is complete.

3.2 The non-cocompact case

In the non-compact case, if P⊂Hnis simple, Theorem 3.2.3 implies that

Pis either a simplex, isometric to P0⊂H4depicted in Figure 3.2.6, or it

contains a pair of disjoint facets in Hn.

Let 4 ≤n≤9. By results of Terragni [77], we know that Γnhas minimal

growth rate among all non-cocompact Coxeter n-simplex groups. Further-

more, we notice that the growth rates of the Coxeter groups Γnsatisfy the

strictly decreasing sequence

τΓ9< τΓ8<··· < τΓ5< τΓ4.(6.2)

Next, we compute the growth rate τ0of the Coxeter group associated with

P0⊂H4, and we see that τΓ4< τ0.

Assume now that Pis neither a simplex nor isometric to the polyhedron P0.

As already said, if Pis simple, then we know that it admits at least one

pair of disjoint facets. In this case, we ﬁnd that the Coxeter diagram Σ of P

contains a subdiagram of type Wi, 1 ≤i≤3, as depicted in Example 5.3.7.

3. Proofs of the two main results

By use of Lemma 5.3.6, we verify that τΓ4< τW1. From Example 5.3.7, and

(6.2), we derive that

τΓn< τW1≤τΓ.

Assume now that Pis non-simple. If Pcontains a pair of disjoint facets in

Hn, we conclude as before. In particular, this ﬁnishes the proof for n= 4, as

we showed in the proof of Corollary 3.2.4 that non-simple Coxeter polyhedra

in H4always admit a pair of parallel facets.

From now on, we assume that n≥5 and that Pdoes not contain any pair

of disjoint facets. Observe the following important inequality reﬁning (6.2)

τΓ5< τΓ3< τΓ4,(6.3)

where Γ3= [6,3,3].

As Pis non-simple, Σ contains an aﬃne subdiagram of rank n−1≥4 which

is made of at least two aﬃne components; see Theorem 1.3.8. By assumption,

each of these aﬃne components diﬀers from e

A1= [∞]. By (3.1), one veriﬁes

that Σ contains at least one connected aﬃne subdiagram of rank 2,3 or 4.

Therefore, as Σ is connected, it contains at least one (possibly non-simple)

extension σof such a diagram.

In Example 5.3.5, we have seen that any simple extension of a connected

aﬃne diagram of rank 2 or 3 encodes a coﬁnite Coxeter simplex group in

IsomH3or IsomH4, whose growth rate is bounded from below by τΓ3or τΓ4.

Therefore, whenever Σ contains an aﬃne diagram of rank r= 2 or 3, we

derive from (6.3) that

τΓn< τΓr+1 ≤τσ≤τΓ.

Assume that the smallest rank for a connected aﬃne diagram in Σ is 4. By

means of (3.1), it follows that n≥7. Recall that any simple extension of

an aﬃne Coxeter diagram of rank 4, up to the four exceptions ∆1,...,∆4,

gives rise to coﬁnite Coxeter simplex group in IsomH5whose growth rate is

bounded from below by τΓ5; see Example 5.3.5. A straightforward computa-

tion shows that τΓ5< τ∆ifor i= 1,...,4. Finally, by Terragni’s Theorem

5.3.1 and by (6.2), we deduce that

τΓn< τΓ5≤τσ≤τΓ.

This ﬁnishes the proof of Theorem 6.2.2.

Chapter 6. Minimal growth rates for hyperbolic Coxeter groups

Final comments. For the proof of Theorem 6.2.1 and Theorem 6.2.2, we

developed a new strategy to identify Coxeter groups of minimal growth rate

in IsomHnfor 4 ≤n≤9. These methods diﬀer completely from the ones

exploited for dimensions 2 and 3, and they are applicable to abstract Coxeter

groups of arbitrarily large rank as well.

However, in the hyperbolic case, as the potential candidates for minimal

growth rate in large dimensions are not simplex groups anymore and have

a comparatively big growth rate value, the ﬁnding of suitable comparison

diagrams causes high computational cost.

In the work of Hild [35], a similar phenomenon showed up in the context

of minimal volume cusped hyperbolic n-orbifolds. He developed a general

method to identify the minimizers, however, he exploited it for dimensions

n≤9, only. The reason there was that the minimizer orbifolds turned out

to be quotients related to the Coxeter simplex groups Γnwhose properties

are most beautiful in various ways.

Only those who attempt the absurd...

will achieve the impossible.

M.C. Escher

Part IV:

Appendices

APPENDIX A

Data for Prokhorov’s formula

In this appendix, we provide the quantities involved in Prokhorov’s formula

for aﬃne components of type e

A, e

Dor e

E. For a component of type e

G2, they

are computed in Example 5.3.8.

In the sequel, we use the representation sn

ij =sn

isn

jfor a speciﬁc root system

Rnwhich is justiﬁed by the following fact. In (4.9), the scalars sn

ij are ex-

pressed in terms of the fundamental weights wn

i, wn

jand the coeﬃcients cn

i, cn

of Rn, however they were originally deﬁned as products ⟨sn

i, sn

j⟩of speciﬁc

vectors sn

i, sn

jdisplayed in explicit form in [70]. In order to simplify notations,

we write sij instead of sn

ij for a ﬁxed root system R=Rn.

⋄For a component of type e

sij =sisj=(j−i)(n+ 1 −(j−i))

2(n+ 1) .

Note that for the proof of Theorem 4.0.2, we need these quantities for e

Anfor

n≤8, only.

⋄For a component of type e











sisj=j−i

2for 2≤i≤j≤n−2

sn+1sj=s1sj=j

4for j≤n−2

s1sn−1=s1jsn=sn−1sn+1 =snsn+1 =n

sn−1sn=s1sn+1 =1

2;sjsn−1=sjsn=n−j

4for j≤n−2

Note that for the proof of Theorem 4.0.2, we need these quantities for e

Dnfor

n≤6.

⋄For a component of type e

E6,e

E7or e

The quantities (sisj)i,j for the components of type e

E6,e

E7and e

E8are given

in the following tables.

s2s3s4s5s6s7

s15

/61

/215

/62

/32

s22

/31

/22

/35

/61

s31

/22

/35

/65

s41

/21 1

s51

/25

s61

s2s3s4s5s6s7s8

s13

/41

/21 1 1 1 1

s25

/81

/25

/83

/47

/87

s31

/33

/415

/41

s41

/213

/23

s51

/215

s61

/21

s73

s2s3s4s5s6s7s8s9

s12

/31

/2111111

s27

/12 1

/22

/35

/617

/64

s31

/23

/415

/43

/27

s41

/213

/225

s51

/213

/22

s61

/213

s71

/21

s81

APPENDIX B

Admissible pairs

In this appendix, we provide all admissible pairs for the cases of the proof

of Theorem 4.0.2 that are not detailed in Chapter 4. Every table contains

admissible pairs of the form {x, yi}and indicates the value ⟨x, yi⟩. Further-

more, when relevant, we provide the Lorentzian products ⟨yi, yj⟩if {yi, yj}is

an admissible pair.

x yi⟨x, yi⟩

(0,0,√3; 1,0,0,0,0,0,0) y1↔(0,√3,√3; 1,1,0,0,0,0,1) 0

y2↔(1,3,3; √3,0,0,√3,0,√3,√3) 0

y3↔(1,3,3; √3,√3,√3,0,√3,0,0) 0

y4↔(√3,√3,0; 1,1,1,0,0,1,1) 0

y5↔(0,√3,√3; 0,0,0,1,0,1,1) -1

y6↔(0,√3,√3; 0,1,1,0,1,0,0) -1

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5y6

y10 0 0 0

y20−√3

y30−√3

y40 0

y50

Table 2.0.1: Admissible pairs {x, yi}for σ∞=e

G2∪e

x yi⟨x, yi⟩

(0,0,√3; 1,0,0,0,0,0,0) y1↔(0,√3,√3; 1,1,0,0,0,0,0) 0

y2↔(1,3,3; √3,0,0,0,√3,√3,√3) 0

y3↔(√3,√3,√3; 0,0,1,1,1,1,0) 0

y4↔(0,√3,√3; 0,0,0,0,1,1,1) -1

The Lorentzian products ⟨yi, yj⟩

y2y3y4

y10 0

y2−√3

y30

Table 2.0.2: Admissible pairs {x, yi}for σ∞=e

G2∪e

x yi⟨x, yi⟩

(0,0,√3; 1,0,0,0,0,0,0) y1↔(0,√3,√3; 1,0,1,0,0,0,0) 0

y2↔(1,3,3; √3,0,0,0,√3,√3,0) 0

y3↔(1,3,3; √3,√3,0,0,0,0,√3) 0

y4↔(0,√3,√3; 0,0,0,0,1,1,0) -1

y5↔(0,√3,√3; 0,1,0,0,0,0,1) -1

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5

y10 0 0 0

Table 2.0.3: Admissible pairs {x, yi}for σ∞=e

G2∪e

Appendix B. Admissible pairs

x yi⟨x, yi⟩

(0,0,√3; 0,0,√3; 0,0,√3; 0,0,√3) (1,0,0; 1,0,0; 1,0,0; 1,0,0) 0

(0,0,3; 0,0,√3; 0,0,√3; 0,0,√3) y1↔(√3,0,0; 0,0,3; 0,0,3; 1,0,0) 0

y2↔(√3,0,0; 0,0,3; 1,0,0; 0,0,3) 0

y3↔(√3,0,0; 1,0,0; 0,0,3; 0,0,3) 0

The Lorentzian products ⟨yi, yj⟩

y2y3

y10 0

y20

Table 2.0.4: Admissible pairs {x, yi}for σ∞=e

G2∪e

x yi⟨x, yi⟩

(0,0,√3; 0,0,√3; 0,0,√3; 1,0,0) (1,0,0; 1,0,0; 1,0,0; 0,0,√3) 0

(1,0,0; 1,0,0; 1,0,0; 0,√3,0) 0

(√3,0,0; √3,0,0; √3,0,0; 1,1,1) 0

(0,0,3; 0,0,√3; 0,0,√3; 1,0,0) y1↔(√3,0,0; 0,0,3; 0,0,3; 0,0,√3) 0

y2↔(√3,0,0; 0,0,3; 0,0,3; 0,√3,0) 0

y3↔(√3,0,0; 0,0,3; 1,0,0; √3,0,0) 0

y4↔(√3,0,0; 1,0,0; 0,0,3; √3,0,0) 0

The Lorentzian products ⟨yi, yj⟩

y3y4

y10 0

y20 0

y30

Table 2.0.5: Admissible pairs {x, yi}for σ∞=e

G2∪e

100

x yi⟨x, yi⟩

(0,0,√3; 0,0,√3; 1,0,0; 1,0,0) y1↔(1,0,0; 1,0,0; 0,0,√3; 0,0,√3) 0

y2↔(1,0,0; 1,0,0; 0,0,√3; 0,√3,0) 0

y3↔(1,0,0; 1,0,0; 0,√3,0; 0,0,√3) 0

y4↔(1,0,0; 1,0,0; 0,√3,0; 0,√3,0) 0

x zi⟨x, zi⟩

(0,0,3; 0,0,√3; 1,0,0; 1,0,0) y1↔(√3,0,0; 0,0,3; 0,0,√3; √3,0,0) 0

y2↔(√3,0,0; 0,0,3; 0,√3,0; √3,0,0) 0

y3↔(√3,0,0; 0,0,3; √3,0,0; 0,0,√3) 0

y4↔(√3,0,0; 0,0,3; √3,0,0; 0,√3,0) 0

y5↔(√3,0,0; 1,0,0; √3,0,0; √3,0,0) 0

The Lorentzian products ⟨yi, yj⟩and ⟨zi, zj⟩

y3y4

y10

y20

z3z4z5

z1000

z2000

z30

z40

Table 2.0.6: Admissible pairs for σ∞=e

G2∪e

A2∪e

101

Appendix B. Admissible pairs

x yi⟨x, yi⟩

(0,0,√3; 1,0,0; 1,0,0; 1,0,0) y1↔(1,0,0; 0,0,√3; 0,0,√3; 0,0,√3) 0

y2= (1,0,0; 0,0,√3; 0,0,√3; 0,√3,0) 0

y3= (1,0,0; 0,0,√3; 0,√3,0; 0,0,√3) 0

y4↔(1,0,0; 0,0,√3; 0,√3,0; 0,√3,0) 0

y5↔(1,0,0; 0,√3,0; 0,0,√3; 0,0,√3) 0

y6↔(1,0,0; 0,√3,0; 0,0,√3; 0,√3,0) 0

y7↔(1,0,0; 0,√3,0; 0,√3,0; 0,0,√3) 0

y8↔(1,0,0; 0,√3,0; 0,√3,0; 0,√3,0) 0

x zi⟨x, zi⟩

(0,0,3,1,0,0,1,0,0,1,0,0) z1↔(1,0,0; 0,0,1; 0,0,1; 0,0,1) 0

z2↔(1,0,0; 0,0,1; 0,0,1; 0,1,0) 0

z3↔(1,0,0; 0,0,1; 0,1,0; 0,0,1) 0

z4↔(1,0,0; 0,0,1; 0,1,0; 0,1,0) 0

z5↔(1,0,0; 0,1,0; 0,0,1; 0,0,1) 0

z6↔(1,0,0; 0,1,0; 0,0,1; 0,1,0) 0

z7↔(1,0,0; 0,1,0; 0,1,0; 0,0,1),0

z8↔(1,0,0; 0,1,0; 0,1,0; 0,1,0) 0

z9↔(√3,0,0; 0,0,√3; √3,0,0; √3,0,0) 0

z10 ↔(√3,0,0; 0,√3,0; √3,0,0; √3,0,0) 0

z11 ↔(√3,0,0; √3,0,0; 0,0,√3; √3,0,0) 0

z12 ↔(√3,0,0; √3,0,0; 0,√3,0; √3,0,0) 0

z13 ↔(√3,0,0; √3,0,0; √3,0,0; 0,0,√3) 0

z14 ↔(√3,0,0; √3,0,0; √3,0,0; 0,√3,0) 0

The Lorentzian products ⟨yi, yj⟩and ⟨zi, zj⟩

y2y3y4y5y6y7y8

y10 0 0 -1

y20 0 -1 0

y30 -1 0

y4-1 0 0

y50

y60

z10 z11 z12 z13 z14

z90 0 0 0

z10 0 0 0 0

z11 0 0

z12 0 0

Table 2.0.7: Admissible pairs for σ∞=e

G2∪e

A2∪e

102

x y ⟨x, y⟩

(0,0,√3; 0,0,√3; 1,0,0,0,0),(0,√3,√3; √3,0,0; 1,1,0,0,1) -1

(√3,0,0; 0,√3,√3; 1,1,0,0,1) -1

(0,0,3; 0,0,√3; 1,0,0,0,0) (√3,0,0; 1,0,0; √3,0,0,0,0) 0

Table 2.0.8: Admissible pairs {x, y}for σ∞=e

G2∪e

x y ⟨x, y⟩

(0,0,3; 1,0,0; 1,0,0,0) (√3,0,0; 0,0,√3; √3,0,0,0,0) 0

(√3,0,0; 0,√3,0; √3,0,0,0,0) 0

Table 2.0.9: Admissible pairs {x, y}for σ∞=e

G2∪e

A2∪e

x y ⟨x, y⟩

(0,0,√3; 1,0,0; 1,0,0,0) y↔(0,0,√3; 1,1,1; 0,0,1,1,1) 0

(0,0,3; 1,0,0; 1,0,0,0) (√3,0,0,0,0,√3,√3,0,0,0,0) 0

(√3,0,0,0,√3,0,√3,0,0,0,0) 0

Table 2.0.10: Admissible pairs {x, y}for σ∞=e

G2∪e

A2∪e

x y ⟨x, y⟩

(0,0,√3,0,0,√3,1,0,0,0,0),(0,√3,√3,√3,0,0,1,1,0,0,0) 0

(√3,0,0,0,√3,√3,1,1,0,0,0) 0

(0,0,3,0,0,√3,1,0,0,0,0) (√3,0,0; 1,0,0; √3,0,0,0,0) 0

Table 2.0.11: Admissible pairs {x, y}for σ∞=e

G2∪e

x y ⟨x, y⟩

(0,0,√3; 1,0,0,0; 1,0,0,0) (1,0,0; 0,0,√3,0; 0,0,√3,0) 0

(0,0,3; 1,0,0,0; 1,0,0,0) (1,0,0; 0,0,1,0; 0,0,1,0) 0

Table 2.0.12: Admissible pairs {x, y}for σ∞=e

G2∪e

A3∪e

103

Appendix B. Admissible pairs

x yi⟨x, yi⟩

(0,0,√3; 1,0,0,0,0,0,0,0) y1↔(0,√3,√3,1,1,0,0,0,0,0,1) 0

y2↔(1,0,0,0,0,0,0,√3,0,0,0) 0

y3↔(1,3,3,√3,0,0,√3,0,0,√3,√3) 0

y4↔(1,3,3,√3,0,√3,√3,0,0,0,√3) 0

y5↔(1,3,3,√3,√3,0,0,0,√3,√3,0) 0

y6↔(1,3,3,√3,√3,√3,0,0,√3,0,0) 0

y7↔(√3,0,0,0,0,0,1,1,1,0,0) 0

y8↔(√3,√3,0,1,1,1,0,0,0,1,1) 0

y9↔(0,√3,√3,0,0,0,1,0,0,1,1) -1

y10 ↔(0,√3,√3,0,0,1,1,0,0,0,1) -1

y11 ↔(0,√3,√3,0,1,0,0,0,1,1,0) -1

y12 ↔(0,√3,√3,0,1,1,0,0,1,0,0) -1

y13 ↔(1,3,3,0,0,0,√3,√3,0,√3,√3) −√3

y14 ↔(1,3,3,0,0,√3,√3,√3,0,0,√3) −√3

y15 ↔(1,3,3,0,√3,0,0,√3,√3,√3,0) −√3

y16 ↔(1,3,3,0,√3,√3,0,√3,√3,0,0) −√3

(0,0,3; 1,0,0,0,0,0,0,0) (1,0,0; 0,0,0,0,1,0,0,0) 0

The Lorentzian products ⟨yi, yj⟩

y3y4y5y6y8y9y10 y11 y12 y13 y14 y15 y16

y10 0 0 0 0 0 0 0

y2−√3−√3−√3−√3-1 -1 -1 -1

y3-1 -1 0 0 0 −√31

y4-1 0 0 −√30 1

y5-1 0 0 −√30 1

y60−√30 0 1

y80 0 0 0

y90 0 0 −√3

y10 0 0 −√30

y11 0−√30

y12 −√30 0

y13 -1 -1

y14 -1

y15 -1

Table 2.0.13: Admissible pairs {x, yi}for σ∞=e

G2∪e

104

x yi⟨x, yi⟩

(0,0,√3; 1,0,0,0,0,0,0,0) y1↔(0,√3,√3; 1,1,0,0,0,0,0,0) 0

y2↔(1,3,3; √3,0,0,0,√3,0,0,√3) 0

y3↔(√3,√3,√3; 0,0,1,1,1,0,0,0) 0

y4↔(0,√3,√3; 0,0,0,0,1,0,0,1) -1

y5↔(√3,√3,0; 0,0,1,0,0,1,1,1) -1

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5

y1-1

y2√3 0

y30 0

Table 2.0.14: Admissible pairs {x, yi}for σ∞=e

G2∪e

x yi⟨x, yi⟩

(0,0,√3; 0,0,0,0,0,0,0,1) y1↔(0,√3,√3; 1,0,0,0,0,0,0,1) 0

y2↔(1,3,3; 0,√3,0,0,0,0,√3,√3) 0

y3↔(√3,√3,0; 1,0,0,0,0,1,0,1) 0

y4↔(√3,√3,√3; 0,0,0,0,1,1,1,0) 0

y5↔(0,√3,√3; 0,1,0,0,0,0,1,0) -1

y6↔(√3,√3,0; 0,1,0,0,1,0,0,0) -1

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5y6

y10 0 1

y20−√30

y3-1 0 0

y40 0

Table 2.0.15: Admissible pairs {x, yi}for σ∞=e

G2∪e

105

Appendix B. Admissible pairs

x y ⟨x, y⟩

(0,0,√3; 0,0,√3; 1,0,0,0,0,0) (0,√3,√3; √3,0,0; 1,1,0,0,0,0) -1

(√3,0,0; 0,√3,√3; 1,1,0,0,0,0) -1

(0,0,3; 0,0,√3; 1,0,0,0,0,0) (√3,0,0; 1,0,0; √3,0,0,0,0,0) 0

Table 2.0.16: Admissible pairs {x, y}for σ∞=e

G2∪e

x y ⟨x, y⟩

(0,0,√3; 1,0,0; 1,0,0,0,0,0) (0,√3,√3; 1,1,1; 0,0,1,0,0,1) -1

(0,0,3; 1,0,0; 1,0,0,0,0,0) (√3,0,0; 0,0,√3; √3,0,0,0,0,0) 0

(√3,0,0; 0,√3,0; √3,0,0,0,0,0) 0

Table 2.0.17: Admissible pairs {x, y}for σ∞=e

G2∪e

A2∪e

x yi⟨x, yi⟩

(0,0,√3; 1,0,0,0; 1,0,0,0,0) y1↔(0,√3,√3; 0,1,1,1; 1,0,0,1,1) -1

y2↔(0,√3,√3; 0,1,1,1; 1,0,1,0,1) -1

y3↔(0,√3,√3; 0,1,1,1; 1,0,1,1,0) -1

y4↔(0,√3,√3; 1,1,0,1; 0,0,1,1,1) -1

y5↔(1,0,0; 0,0,√3,0; 0,0,0,0,√3) 0

y6↔(1,0,0; 0,0,√3,0; 0,0,0,√3,0) 0

y7↔(1,0,0; 0,0,√3,0; 0,0,√3,0,0) 0

(0,0,3; 1,0,0,0; 1,0,0,0,0) (1,0,0; 0,0,1,0; 0,0,0,0,1) 0

(1,0,0; 0,0,1,0; 0,0,0,1,0) 0

(1,0,0; 0,0,1,0; 0,0,1,0,0) 0

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5y6y7

y1−√3

y2−√3

y3−√3

y4−√3−√3−√3

Table 2.0.18: Admissible pairs {x, yi}for σ∞=e

G2∪e

A3∪e

106

x yi⟨x, yi⟩

(0,0√3; 1,0,0,0,0,0,0,0,0) y1↔(0,√3,√3; 1,1,0,0,0,0,0,0,0) 0

y2↔(1,0,0; 0,0,0,0,0,0,0,√3,0) 0

y3↔(1,0,0; 0,0,0,0,0,0,√3,0,0) 0

y4↔(1,3,3; 0,0,0,0,√3,0,0,√3,√3) −√3

y5↔(1,3,3; 0,0,0,0,√3,0,√3,0,√3) −√3

y6↔(1,3,3,; 0,0,√3,0,0,0,√3,√3,0) −√3

y7↔(1,3,3; √3,0,0,0,√3,0,0,0,√3) 0

y8↔(1,3,3; √3,0,√3,0,0,0,0,√3,0) 0

y9↔(1,3,3; √3,0,√3,0,0,0,√3,0,0) 0

y10 ↔(√3,0,0; 0,0,0,0,0,1,0,1,0) 0

y11 ↔(√3,0,0; 0,0,0,0,0,1,1,0,0) 0

y12 ↔(√3,√3,0; 1,1,0,0,0,0,0,1,1) 0

y13 ↔(√3,√3,0; 1,1,0,0,0,0,1,0,1) 0

y14 ↔(√3,√3,0; 1,1,0,1,0,0,0,0,0) 0

y15 ↔(√3,√3,√3; 0,0,1,0,0,1,0,1,1) 0

y16 ↔(√3,√3,√3; 0,0,1,0,0,1,1,0,1) 0

y17 ↔(√3,√3,√3; 0,0,1,1,1,0,0,0,0) 0

y18 ↔(√3,√3,√3; 0,1,0,0,1,0,1,1,0) 0

(0,0,3; 1,0,0,0,0,0,0,0,0) (1,0,0; 0,0,0,0,0,0,0,1,0) 0

(1,0,0; 0,0,0,0,0,0,1,0,0) 0

The Lorentzian products ⟨yi, yj⟩

y4y5y6y7y8y9y14 y15 y16 y17 y18

y10 0 0

y2-1 -1 0 −√30

y3-1 -1 0 −√30

y4-1 -1 0 −√30

y5-1 -1 0 −√30

y6-1 -1 0 0 −√30

y7-1 −√3−√3

y9−√3−√3

y10 -1 -1

y11 -1 -1

y14 -1

y15 0

y16 0

y17 0

Table 2.0.19: Admissible pairs {x, yi}for σ∞=e

G2∪e

107

Appendix B. Admissible pairs

x yi⟨x, yi⟩

(0,0√3; 0,0,0,0,0,0,0,0,1) y1↔(0,√3,√3; 0,0,0,0,0,0,0,1,1),0

y2↔(0,√3,√3; 0,1,0,0,0,0,0,0,0),-1

y3↔(1,3,3; 0,√3,0,0,0,0,0,0,√3),0

y4↔(√3,√3,0; 0,0,0,0,1,0,0,0,0) -1

y5↔(√3,√3,0; 1,0,0,0,0,0,0,1,1) 0

y6↔(√3,√3,√3; 1,0,1,0,0,0,0,0,0) 0

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5y6

y10 0 -1

y20 0

y30 0 −√3

y40 0

y5-1

Table 2.0.20: Admissible pairs {x, yi}for σ∞=e

G2∪e

x y ⟨x, y⟩

(0,0,3; 1,0,0,0,0; 1,0,0,0,0) (1,0,0; 0,0,0,0,1; 0,0,0,0,1) 0

(1,0,0; 0,0,0,0,1; 0,0,0,1,0) 0

(1,0,0; 0,0,0,0,1; 0,0,1,0,0) 0

(1,0,0; 0,0,0,1,0; 0,0,0,0,1) 0

(1,0,0; 0,0,0,1,0; 0,0,0,1,0) 0

(1,0,0; 0,0,0,1,0; 0,0,1,0,0) 0

(1,0,0; 0,0,1,0,0; 0,0,0,0,1) 0

(1,0,0; 0,0,1,0,0; 0,0,0,1,0) 0

(1,0,0; 0,0,1,0,0; 0,0,1,0,0) 0

Table 2.0.21: Admissible pairs {x, y}for σ∞=e

G2∪e

D4∪e

108

x y ⟨x, y⟩

(0,0,√3; 1,0,0,0,0; 1,0,0,0,0) y1↔(1,0,0; 0,0,0,0,√3; 0,0,0,0,√3) 0

y2↔(1,0,0; 0,0,0,0,√3; 0,0,0,√3,0) 0

y3↔(1,0,0; 0,0,0,0,√3; 0,0,√3,0,0) 0

y4↔(1,0,0; 0,0,0,√3,0; 0,0,0,0,√3) 0

y5↔(1,0,0; 0,0,0,√3,0; 0,0,0,√3,0) 0

y6↔(1,0,0; 0,0,0,√3,0; 0,0,√3,0,0) 0

y7↔(1,0,0; 0,0,√3,0,0; 0,0,0,0,√3) 0

y8↔(1,0,0; 0,0,√3,0,0; 0,0,0,√3,0) 0

y9↔(1,0,0; 0,0,√3,0,0; 0,0,√3,0,0) 0

y10 ↔(0,√3,√3; 0,0,1,1,1; 1,1,0,0,0) -1

y11 ↔(0,√3,√3; 0,1,0,0,1; 1,0,0,1,1) -1

y12 ↔(0,√3,√3; 0,1,0,0,1; 1,0,1,0,1) -1

y13 ↔(0,√3,√3; 0,1,0,0,1; 1,0,1,1,0) -1

y14 ↔(0,√3,√3; 0,1,0,1,0; 1,0,0,1,1) -1

y15 ↔(0,√3,√3; 0,1,0,1,0; 1,0,1,0,1) -1

y16 ↔(0,√3,√3; 0,1,0,1,0; 1,0,1,1,0) -1

y17 ↔(0,√3,√3; 0,1,1,0,0; 1,0,0,1,1) -1

y18 ↔(0,√3,√3; 0,1,1,0,0; 1,0,1,0,1) -1

y19 ↔(0,√3,√3; 0,1,1,0,0; 1,0,1,1,0) -1

y20 ↔(0,√3,√3; 1,0,0,1,1; 0,1,0,0,1) -1

y21 ↔(0,√3,√3; 1,0,0,1,1; 0,1,0,1,0) -1

y22 ↔(0,√3,√3; 1,0,0,1,1; 0,1,1,0,0) -1

y23 ↔(0,√3,√3; 1,0,1,0,1; 0,1,0,0,1) -1

y24 ↔(0,√3,√3; 1,0,1,0,1; 0,1,0,1,0) -1

y25 ↔(0,√3,√3; 1,0,1,0,1; 0,1,1,0,0) -1

y26 ↔(0,√3,√3; 1,0,1,1,0; 0,1,0,0,1) -1

y27 ↔(0,√3,√3; 1,0,1,1,0; 0,1,0,1,0) -1

y28 ↔(0,√3,√3; 1,0,1,1,0; 0,1,1,0,0) -1

y29 ↔(0,√3,√3; 1,1,0,0,0; 0,0,1,1,1) -1

Table 2.0.22: Admissible pairs {x, yi}for σ∞=e

G2∪e

D4∪e

D4(continued)

109

Appendix B. Admissible pairs

The Lorentzian products ⟨yi, yj⟩

y4y5y6y7y8y9y10 y11 y12 y13 y14 y15 y16 y17

y11 1 1 1 −√3−√3−√3−√3−√3

y21 1 1 1 −√3−√3−√3−√3−√3

y31 1 1 1 −√3−√3−√3−√3

y41 1 −√3−√3−√3−√3

y51 1 −√3−√3−√3−√3−√3

y61 1 −√3−√3−√3−√3

y7−√3−√3−√3−√3−√3

y8−√3−√3−√3−√3−√3

y9−√3−√3−√3−√3−√3−√3

y18 y19 y20 y21 y22 y23 y24 y25 y26 y27 y28 y29

y1−√3−√3−√3−√3−√3−√3−√3

y2−√3−√3−√3−√3−√3−√3−√3

y3−√3−√3−√3−√3−√3−√3−√3−√3

y4−√3−√3−√3−√3−√3−√3−√3

y5−√3−√3−√3−√3−√3−√3−√3

y6−√3−√3−√3−√3−√3−√3−√3−√3

y7−√3−√3−√3−√3−√3−√3−√3

y8−√3−√3−√3−√3−√3−√3−√3

y9−√3−√3−√3−√3−√3−√3

y20 y21 y22 y23 y24 y25 y26 y27 y28 y29

y10 0

y11 0

y12 0

y13 0

y14 0

y15 0

y16 0

y17 0

y18 0

y19 0

Table 2.0.23: Lorentzian products ⟨yi, yj⟩for σ∞=e

G2∪e

D4∪e

110

x yi⟨x, yi⟩

(0,0√3; 1,0,0; 1,0,0,0,0,0,0) y1↔(1,0,0; 0,0,√3; 0,0,0,0,0,0,√3)

y2= (1,0,0; 0,0,√3; 0,0,0,0,0,√3,0)

y3= (1,0,0; 0,√3,0; 0,0,0,0,0,0,√3)

y4= (1,0,0; 0,√3,0; 0,0,0,0,0,√3,0)

y5= (0,√3,√3; 1,1,1; 0,0,0,1,0,0,0)

y6= (0,3,3; √3,√3,√3; √3,0,0,0,0,√3,√3)

(0,0,3; 1,0,0; 1,0,0,0,0,0,0) (1,0,0; 0,0,1; 0,0,0,0,0,0,1)

(1,0,0; 0,0,1; 0,0,0,0,0,1,0)

(1,0,0; 0,1,0; 0,0,0,0,0,0,1)

(1,0,0; 0,1,0; 0,0,0,0,0,1,0)

(√3,0,0; 0,0,√3; √3,0,0,0,0,0,0)

(√3,0,0; 0,√3,0; √3,0,0,0,0,0,0)

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5y6

y10−√3

y20−√3

y3−√3

y4−√3

y50

Table 2.0.24: Admissible pairs {x, yi}for σ∞=e

G2∪e

A2∪e

111

Appendix B. Admissible pairs

x yi⟨x, y⟩

(0,0√3; 0,0,√3; 1,0,0,0,0,0,0), y1↔(1,0,0; 1,0,0; 0,0,0,0,0,0,√3) 0

y2↔(1,0,0; 1,0,0; 0,0,0,0,0,√3,0) 0

y3↔(√3,0,0; √3,0,0; 0,0,0,0,1,1,0) 0

y4↔(√3,0,0; √3,0,0; 0,1,0,0,0,0,1) 0

y5↔(0,√3,√3; √3,0,0; 1,0,1,0,0,0,0) -1

y6↔(√3,0,0; 0,√3,√3; 1,0,1,0,0,0,0) -1

(0,0,3; 0,0,√3; 1,0,0,0,0,0,0) (√3,0,0; 1,0,0; √3,0,0,0,0,0,0) 0

The Lorentzian products ⟨yi, yj⟩

y2y3y4y5y6

y1√3√3

y2√3√3

Table 2.0.25: Admissible pairs {x, yi}for σ∞=e

G2∪e

x y ⟨x, y⟩

(0,0,√3; 0,0,√3; 0,0,0,0,0,0,0,0,1) (√3,0,0; √3,0,0; 0,1,0,0,0,0,0,0,0) 0

(0,√3,√3; √3,0,0; 0,0,0,0,0,0,0,1,1) -1

(√3,0,0; 0,√3,√3; 0,0,0,0,0,0,0,1,1) -1

(0,0,3; 0,0,√3; 0,0,0,0,0,0,0,0,1) (√3,0,0; 1,0,0; 0,0,0,0,0,0,0,0,√3) 0

Table 2.0.26: Admissible pairs {x, y}for σ∞=e

G2∪e

112

x yi⟨x, yi⟩

a=√3y1↔(1,0,3; √3,0,0,0,0,0,0,0,0; √3,0,0,0,0,0,0,0,0) 0

y2↔(√3,0,0; 0,0,0,0,0,0,1,0,0; 0,1,0,0,0,0,0,0,0) 0

y3↔(√3,0,0; 0,1,0,0,0,0,0,0,0; 0,0,0,0,0,0,1,0,0) 0

y4↔(√3,0,√3; 0,0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0) 0

(√3,√3,√3; 0,0,0,0,0,0,1,1,1; 1,0,0,0,0,0,1,0,1) 0

(√3,√3,√3; 0,1,0,0,0,0,0,1,1; 0,1,0,0,0,0,0,1,1) 0

(√3,√3,√3; 1,0,0,0,0,0,1,0,1; 0,0,0,0,0,0,1,1,1) 0

(0,√3,√3; 0,0,0,0,0,0,0,1,1; 0,0,0,0,0,0,1,0,0) -1

(0,√3,√3; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,1) -1

(0,√3,√3; 1,0,0,0,0,0,0,0,1; 1,0,0,0,0,0,0,0,1) -1

(√3,√3,0; 1,0,0,0,0,0,0,1,1; 1,0,0,0,0,0,0,1,1) -1

(√3,√3,√3; 0,0,0,0,0,0,1,1,1; 0,0,0,1,0,0,0,0,0) -1

(√3,√3,√3; 0,0,0,0,0,1,0,1,0; 0,0,0,0,0,1,0,1,0) -1

(√3,√3,√3; 0,0,0,0,0,1,0,1,0; 1,1,0,0,0,0,0,0,1) -1

(√3,√3,√3; 0,0,0,0,1,0,0,0,1; 0,0,1,0,0,0,0,1,0) -1

(√3,√3,√3; 0,0,0,1,0,0,0,0,0; 0,0,0,0,0,0,1,1,1) -1

(√3,√3,√3; 0,0,1,0,0,0,0,1,0; 0,0,0,0,1,0,0,0,1) -1

(√3,√3,√3; 0,1,0,0,0,0,0,1,1; 1,0,0,0,0,1,0,0,0) -1

(√3,√3,√3; 0,1,0,0,0,0,1,0,0; 1,0,0,0,0,0,1,0,1) -1

(√3,√3,√3; 1,0,0,0,0,0,1,0,1; 0,1,0,0,0,0,1,0,0) -1

(√3,√3,√3; 1,0,0,0,0,1,0,0,0; 0,1,0,0,0,0,0,1,1) -1

(√3,√3,√3; 1,1,0,0,0,0,0,0,1; 0,0,0,0,0,1,0,1,0) -1

(√3,√3,√3; 1,1,0,0,0,0,0,0,1; 1,1,0,0,0,0,0,0,1) -1

(1,3,3; 0,0,0,0,0,0,√3,0,√3; √3,0,0,0,0,0,0,√3,0) −√3

(1,3,3; 0,√3,0,0,0,0,0,0,√3; 0,√3,0,0,0,0,0,0,√3) −√3

(1,3,3; √3,0,0,0,0,0,0,√3,0; 0,0,0,0,0,0,√3,0,√3) −√3

a= 3 (1,0,3; 1,0,0,0,0,0,0,0,0; 1,0,0,0,0,0,0,0,0) 0

The Lorentzian products ⟨yi, yj⟩

y2y3y4

y1−√3−√30

y20 0

y30

Table 2.0.27: Admissible pairs {x, yi}for σ∞=e

G2∪e

E8∪e

E8and where

x↔(0,0, a; 0,0,0,0,0,0,0,0,1; 0,0,0,0,0,0,0,0,1)

113

APPENDIX C

Normal vectors and Vinberg form

In this appendix, we provide the normal vectors of the ADEG-polyhedron

P⋆and the ADE-polyhedron P2in H9. For the proof of Corollary 4.3.1, we

construct the Vinberg forms of their associated Coxeter groups. Very useful

in this context is the material with the worked examples as presented in

[32, 33]. We start by a short description of the procedure.

⋄Establishing the Vinberg form

Let Γ(P) be a hyperbolic Coxeter group of rank N.

Let e1, . . . , eNbe the unit outer normal vectors of its Coxeter polyhedron

P⊂Hn. For G= (gij)i,j its Gram matrix, consider all cycles of 2G, that is,

ci1i2...ik= 2kgi1i2···gik−1ikgiki1for i1, . . . , ik∈ {1, . . . , N}, k ≥2 ;

see Deﬁnition 1.3.19. The ﬁeld of deﬁnition Kof Γ is the ﬁeld Q({ci1i2...ik})

of all cycles of 2G.

Deﬁne the vectors

v1:= 2e1and vi1i2...ik:= 2kgi1i2···gik−1ikeik(C.1)

by multiplication of the vectors e1, . . . , eNwith partial cycles. Then, the

K-vector space Vspanned by the vectors vi1i2...ikhas dimension n+ 1, and

the restriction of the Lorentzian product on Vyields a quadratic form qof

signature (n, 1), called the Vinberg form of Γ.

Next, we apply this procedure for the arithmetic groups Γ⋆and Γ(P2), where

the ﬁeld K=Q; see Theorem 1.3.20.

114

⋄For the Coxeter polyhedron P⋆

We give below the Coxeter diagram of P⋆⊂H9where the nodes are indexed

by its unit outer normal vectors.

e12 e1e2

e13 e3e4

e14 e5e6

e10 e9e8

e11 e7

The vectors e1, . . . , e14 ∈R10 and their Gram matrix G⋆are given as follows.

e1= (1,0,0,0,0,0,0,0,0,0)

e2= (−1

2,√3

2,0,0,0,0,0,0,0,0)

e3= (0,0,1,0,0,0,0,0,0,0)

e4= (0,0,−1

2,√3

2,0,0,0,0,0,0)

e5= (0,0,0,0,1,0,0,0,0,0)

e6= (0,0,0,0,−1

2,√3

2,0,0,0,0)

e7= (0,−1

√3,0,−1

√3,0,0,−1,1)

e8= (0,0,0,0,0,0,1

2,0,1,−1

e9= (0,0,0,0,0,0,0,−1,−1,1)

e10 = (0,0,0,0,0,0,−1

2,√3

2,1

2,−1

e11 = (0,0,0,0,0,0,1,0,−1,1)

e12 = (−√3

2,−1

2,0,0,0,0,−3+√3

6,−1

2√3,3−2√3

6,−1+√3

e13 = (0,0,−√3

2,−1

2,0,0,−3+√3

6,−1

2√3,3−2√3

6,−1+√3

e14 = (0,0,0,0,−√3

2,−1

2,−3+√3

6,−1

2√3,3−2√3

6,−1+√3







1−1

/20 0 0 0 0 0 0 0 0 −√3

/20 0

−1

/210000−1

/20 0 0 0 0 0 0

0 0 1 −1

/20 0 0 0 0 0 0 0 −√3

/20

0 0 −1

/2100−1

/20 0 0 0 0 0 0

00001−1

/20 0 0 0 0 0 0 −√3

0 0 0 0 −1

/21−1

/20 0 0 0 0 0 0

0−1

/20−1

/21−1

/20 0 0 0 0 0

000000−1

/21−1

/20 0 0 0 0

0000000−1

/21−√3

/20 0 0 0

0 0 0 0 0 0 0 0 −√3

/21−1

/20 0 0

000000000−1

/21−1

/2−1

−√3

/20 0 0 0 0 0 0 0 0 −1

/21 0 0

0 0 −√3

/20 0 0 0 0 0 0 −1

/20 1 0

0 0 0 0 −√3

/20 0 0 0 0 −1

/20 0 1







115

Appendix C. Normal vectors and Vinberg form

Let V(Γ⋆) be the Q-vector space of dimension 10 spanned by the vectors as

given by (C.1). The following vectors form a basis of V(Γ⋆).

v1= 2 ·e1= (2,0,0,0,0,0,0,0,0,0)

v2= 22g11g12 ·e2= (1,−√3,0,0,0,0,0,0,0,0)

v3= 23g11g12g27 ·e7= (0,−2

√3,0,−2

√3,0,0,−2,2)

v4= 24g11g12g27g74 ·e4= (0,0,1,−√3,0,0,0,0,0,0)

v5= 26g11g12g27g74g43g3,13 ·e13 = (0,0,3,√3,0,0,3−√3

√3,1,−3+2√3

√3,√3−3)

v6= 27g11g12g27g74g43g3,13g13,11 ·e11 = (0,0,0,0,0,0,2√3,0,−2√3,2√3)

v7= 28g11g12g27g74g43g3,13g13,11g11,12 ·e12 = (0,0,0,0,0,0,−1,0,−2,1)

v8= 24g11g12g27g78 ·e8= (0,0,0,0,1,−√3,0,0,0,0)

v9= 25g11g12g27g78g89 ·e9= (0,0,0,0,0,0,0,−2,−2,2)

v10 = 24g11g12g27g76 ·e6= (3,√3,0,0,0,0,3−√3

√3,1,3−2√3

√3,−3 + √3)

Therefore, the matrix representing the Vinberg form q⋆of Γ⋆is as follows.

(⟨vi, vj⟩)ij =







4 2 0 0 0 0 0 0 0 6

2 4 2 0 0 0 0 0 0 0

0 2 4 2 0 0 2 2 0 0

0 0 2 4 0 0 0 0 0 0

000012 6 000 0

0000 6 12000 6

0 0 2 0 0 0 4 0 2 0

0 0 2 0 0 0 0 4 0 0

0 0 0 0 0 0 2 0 4 0

6000 0 6 00012







With the help of Mathematica [85], we derive that the diagonal form of q⋆is

given by

<1,1,2,3,3,6,6,10,10,−2> .

116

⋄For the Coxeter polyhedron P2

We give below the Coxeter diagram of P2⊂H9where the nodes are indexed

by its outer normal vectors.

e10

e12

e11

The vectors e1, . . . , e12 ∈R10 and their Gram matrix are given as follows.

e1= (1,0,0,0,0,0,0,0,0,0)

e2= (−1

2,√3

2,0,0,0,0,0,0,0,0)

e3= (0,−1

√3,0,0,0,0,0,0,−√2

√3,0)

e4= (0,0,0,0,0,0,0,−√5

2√2,√3

2√2,0)

e5= (0,0,1,0,0,0,−√3

√5,√2

√5,0,1)

e6= (0,0,1,0,0,−√7

2√3,√5

2√3,0,0,1)

e7= (0,0,1,0,−2

√7,√3

√7,0,0,0,1)

e8= (0,0,0,−3

4,√7

4,0,0,0,0,0)

e9= (−1

2,−1

2√3,0,3

4,1

4√7,1

2√21 ,1

2√15 ,1

2√10 ,1

2√6,0)

e10 = (−1

2,−1

2√3,16

6,−2

3,−2

√7,−5

3√21 ,1

3√15 ,√2

√5,√2

√3,5

e11 = (0,0,−5

6,−1

2√7,4

3√21 ,1

3√15 ,√2

√5,0,−1

e12 = (0,0,1

6,−5

12 ,−5

4√7,−11

3√21 ,−√5

3√3,0,0,1







1−1

/20 0 0 0 0 0 −1

/2−1

/20 0

−1

/21−1

/2000000000

0−1

/21−1

/20 0 0 0 0 −1

/20 0

0 0 −1

/21−1

/20 0 0 0 0 −1

/20

0 0 0 −1

/21−1

/20 0 0 0 0 0

0 0 0 0 −1

/21−1

/20 0 0 −1

/20

0 0 0 0 0 −1

/21−1

/20 0 0 −1

0 0 0 0 0 0 −1

/21−1

/20 0 0

−1

/20 0 0 0 0 0 −1

/21 0 0 −1

−1

/20−1

/2000000100

0 0 0 −1

/20−1

/20 0 0 0 1 0

0 0 0 0 0 0 −1

/20−1

/20 0 1







117

Appendix C. Normal vectors and Vinberg form

Let V(Γ(P2)) be the Q-vector space of dimension 10 spanned by the vectors

as given by (C.1). The following vectors form a basis of V(Γ(P2)).

v1= (2,0,0,0,0,0,0,0,0,0)

v2= (1,−√3,0,0,0,0,0,0,0,0)

v3= (0,−2

√3,0,0,0,0,0,0,−2√2

√3,0)

v4= (0,0,0,0,0,0,0,√5

√2,−√3

√2,0)

v5= (0,0,2,0,0,0,−2√3

√5,2√2

√5,0,2)

v6= (0,0,−2,0,0,√7

√3,−√5

√3,0,0,−2)

v7= (0,0,2,0,−4

√3),2√3

√7,0,0,0,2)

v8= (0,0,0,3

2,−7

2,0,0,0,0,0)

v9= (1,1

√3,−13

3,4

√7,10

3√21 ,−2

3√15 ,−2√2

√5,−2√2

√3,−5)

v10 = (−1,−1

√3,0,3

2,1

2√7,1

√21 ,1

√15 ,1

√10 ,1

√6,0)

The matrix representing the Vinberg form q2of Γ(P2) is given below.

(⟨vi, vj⟩)ij =







4 20000002-2

2 42000000 0

0 24200002 0

0 02420000 0

0 00242000 0

0 00024200 0

0 00002420 0

0 00000240 2

2 02000004 0

-200006020 4







With the help of Mathematica [85], we derive that the diagonal form of q2is

given by

<1,1,3,6,7,10,10,15,21,−10 > .

118

APPENDIX D

The two reprints

This appendix consists of reprints of the following two articles.

•N. Bredon, R. Kellerhals, Hyperbolic Coxeter groups and minimal growth

rates in dimensions four and ﬁve, Groups Geom. Dyn. 16 (2022), 725–741.

•N. Bredon, Hyperbolic Coxeter groups of minimal growth rates in higher

dimensions, Canad. Math. Bull. 66 (2023), 232–242.

119

Groups Geom. Dyn. 16 (2022), 725–741

DOI 10.4171/GGD/663

Published by EMS Press

This work is licensed under a CC BY 4.0 license

Hyperbolic Coxeter groups and minimal growth rates

in dimensions four and ﬁve

Naomi Bredon and Ruth Kellerhals

Abstract. For small n, the known compact hyperbolic n-orbifolds of minimal volume are intim-

ately related to Coxeter groups of smallest rank. For nD2and 3, these Coxeter groups are given by

the triangle group Œ7; 3 and the tetrahedral group Œ3; 5; 3, and they are also distinguished by the fact

that they have minimal growth rate among all cocompact hyperbolic Coxeter groups in Isom Hn,

respectively. In this work, we consider the cocompact Coxeter simplex group G4with Coxeter sym-

bol Œ5; 3; 3; 3 in Isom H4and the cocompact Coxeter prism group G5based on Œ5; 3; 3; 3; 3 in

Isom H5. Both groups are arithmetic and related to the fundamental group of the minimal volume

arithmetic compact hyperbolic n-orbifold for nD4and 5, respectively. Here, we prove that the

group Gnis distinguished by having smallest growth rate among all Coxeter groups acting cocom-

pactly on Hnfor nD4and 5, respectively. The proof is based on combinatorial properties of

compact hyperbolic Coxeter polyhedra, some partial classiﬁcation results and certain monotonicity

properties of growth rates of the associated Coxeter groups.

In memoriam Ernest B. Vinberg

1. Introduction

Let Hndenote the real hyperbolic n-space and Isom Hnits isometry group. A hyper-

bolic Coxeter group GIsom Hnof rank Nis a coﬁnite discrete group generated by N

reﬂections with respect to hyperplanes in Hn. Such a group corresponds to a ﬁnite volume

Coxeter polyhedron PHnwith Nfacets, which in turn is a convex polyhedron all of

whose dihedral angles are of the form 

kfor an integer k2. Hyperbolic Coxeter groups

are geometric realisations of abstract Coxeter systems .W; S/ consisting of a group W

with a ﬁnite set Sof generators satisfying the relations s2D1and .ss0/mss0D1where

mss0Dms0s2 ¹2; 3; : : : ; 1º for s6D s0. For small rank N, the group Wis characterised

most conveniently by its Coxeter symbol or its Coxeter graph.

Hyperbolic Coxeter groups are not only characterised by a simple presentation but

they are also distinguished in other ways. For example, for small n, they appear as fun-

damental groups of smallest volume orbifolds OnDHn= where Isom Hnis a

discrete subgroup; see, e.g., [1,2,7,15,20,27]. In particular, for nD2and 3, the compact

2020 Mathematics Subject Classiﬁcation. Primary 20F55; Secondary 26A12, 22E40, 11R06.

Keywords. Coxeter group, hyperbolic polyhedron, disjoint facets, growth rate.

N. Bredon and R. Kellerhals 726

hyperbolic n-orbifold of minimal volume is the quotient of Hnby a Coxeter group of

smallest rank and given by the triangle group Œ7; 3 and the Z2-extension of the tetrahedral

group Œ3; 5; 3. For nD4and 5, and by restricting to the arithmetic context, the compact

hyperbolic n-orbifold of minimal volume is the quotient of Hnby the 4-simplex group

Œ5; 3; 3; 3 and by the Coxeter 5-prism group based on Œ5; 3; 3; 3; 3, respectively.

In parallel to volume we are interested in the spectrum of small growth rates of hyper-

bolic Coxeter groups GD.W; S /. In general, the growth series fS.t/ of a Coxeter system

.W; S/ is given by

fS.t/ D1CX

k1

aktk;

where ak2Zis the number of elements w2Wwith S-length k. The series fS.t/ can be

computed by Steinberg’s formula

fS.t1/DX

WT<W

jWTj<1

.1/jTj

fT.t/ ;

where WT,TS, is a ﬁnite Coxeter subgroup of W, and where W¿D ¹1º. In particular,

fS.t/ is a rational function that can be expressed as the quotient of coprime monic poly-

nomials p.t/; q.t/ 2ZŒt  of equal degree. For cocompact hyperbolic Coxeter groups, the

series fS.t/ is inﬁnite and has radius of convergence R < 1 which can be identiﬁed with

the real algebraic integer given by the smallest positive root of the denominator polyno-

mial q.t/. The growth rate GD.W;S/ is deﬁned by

GDlim sup

k!1

pak;

and Gcoincides with the inverse of the radius of convergence Rof fS.t/. In contrast to

the ﬁnite and afﬁne cases, hyperbolic Coxeter groups are of exponential growth.

In [16] and [21], it is shown that the triangle group Œ7; 3 and the tetrahedral group

Œ3; 5; 3 have minimal growth rate among all cocompact hyperbolic Coxeter groups in

IsomHnfor nD2and 3, respectively. These results have an interesting number theoretical

component since the growth rate of any Coxeter group acting cocompactly on Hnfor

nD2and 3is either a quadratic unit or a Salem number, that is, is a real algebraic

integer ˛ > 1 whose inverse is a conjugate of ˛, and all other conjugates lie on the unit

circle. In particular, the growth rate Œ7;3 equals the smallest known Salem number, and it

is given by Lehmer’s number ˛L1:17628 with minimal polynomial

L.t/ Dt10 Ct9t7t6t5t4t3CtC1:

The constant ˛Lplays an important role in the strong version of Lehmer’s problem about

a universal lower bound for Mahler measures of non-zero non-cyclotomic irreducible

integer polynomials; see [32].

Hyperbolic Coxeter groups and minimal growth rates in dimensions 4 and 5 727

The proof in [21] of the two results above is based on the fact that for nD2and 3the

rational function fS.t/ comes with an explicit formula in terms of the exponents of the

Coxeter group GD.W; S / Isom Hn.

For dimensions n4, however, there are only a few structural results, and closed for-

mulas for growth functions do not exist in general. In this work, we establish the following

results for nD4and 5by developing a new proof strategy.

Theorem A. Among all Coxeter groups acting cocompactly on H4, the Coxeter simplex

group Œ5; 3; 3; 3 has minimal growth rate, and as such it is unique.

The cocompact Coxeter prism group based on Œ5; 3; 3; 3; 3 in Isom H5was ﬁrst dis-

covered by Makarov [26] and arises as the discrete group generated by the reﬂections in

the compact straight Coxeter prism Mwith base Œ5; 3; 3; 3. More concretely, the prism M

is the truncation of the (inﬁnite volume) Coxeter 5-simplex Œ5; 3; 3; 3; 3 by means of the

polar hyperplane associated to its ultra-ideal vertex characterised by the vertex simplex

Œ5; 3; 3; 3. Our second result can be stated as follows.

Theorem B. Among all Coxeter groups acting cocompactly on H5, the Coxeter prism

group based on Œ5; 3; 3; 3; 3 has minimal growth rate, and as such it is unique.

The work is organised as follows.

In Section 2.1, we provide the necessary background about hyperbolic Coxeter poly-

hedra, their reﬂection groups and the characterisation by means of the Vinberg graph and

the Gram matrix. We present the relevant classiﬁcation results for families of Coxeter

polyhedra with few facets due to Esselmann, Kaplinskaja and Tumarkin. Of particular

importance is the structural result, presented in Theorem 1and due to Felikson and Tumar-

kin, about the existence of non-intersecting facets of a compact Coxeter polyhedron.

In Section 2.2, we consider abstract Coxeter systems with their Coxeter graphs and

Coxeter symbols and introduce the notions of growth series and growth rates. Another

important ingredient is the growth monotonicity result of Terragni as given in Theorem 2.

The proofs of our results are presented in Section 3. The proof of Theorem Ais based

on a simple growth rate comparison argument and serves as an inspiration how to attack

the proof of Theorem B. To this end, we establish Lemma 1and Lemma 2about the com-

parison of growth rates of certain Coxeter groups of rank 4. Then, we consider compact

Coxeter polyhedra in H5in terms of the number N6of their facets. Since compact

hyperbolic Coxeter n-simplices exist only for n4, we look at compact Coxeter poly-

hedra PH5with ND7,ND8and N9facets, respectively. Certain classiﬁcation

results help us dealing with the cases ND7and 8while for N9, we look for particular

subgraphs in the Coxeter graph of Pand conclude by means of Lemma 1, Lemma 2and

Theorem 2.

N. Bredon and R. Kellerhals 728

2. Hyperbolic Coxeter polyhedra and growth rates

2.1. Hyperbolic Coxeter polyhedra and their reﬂection groups

Denote by Hnthe standard hyperbolic n-space realised by the upper sheet of the hyper-

boloid in RnC1according to

HnD®x2RnC1jqn;1.x/ Dx2

1C  C x2

nx2

nC1D 1; xnC1> 0¯:

A hyperbolic hyperplane His the intersection of a vector subspace of dimension nwith

Hnand can be represented as the Lorentz-orthogonal complement HDeLby means of a

vector eof (normalised) Lorentzian norm qn;1.e/ D1. The isometry group IsomHnof Hn

is given by the group OC.n; 1/ of positive Lorentzian matrices leaving the bilinear form

hx; yin;1 associated to qn;1 and the upper sheet invariant. It is well known that OC.n; 1/ is

generated by linear reﬂections rDrHWx7! x2he; xin;1 ewith respect to hyperplanes

HDeL; see [3, Section A.2].

A hyperbolic n-polyhedron PHnis the non-empty intersection of a ﬁnite number

NnC1of half-spaces H

ibounded by hyperplanes Hiall of whose normal unit

vectors eiare directed outwards with respect to P, say. A facet of Pis the intersection of

Pwith one of the hyperplanes Hi,1iN. A polyhedron is a Coxeter polyhedron if

all of its dihedral angles are of the form 

kfor an integer k2.

In this work, we suppose that Pis a compact hyperbolic Coxeter polyhedron so that

Pis the convex hull of ﬁnitely many points in Hn. In particular, Pis simple since all

dihedral angles of Pare less than or equal to 

2. As a consequence, each vertex pof Pis

the intersection of nhyperplanes bounding Pand characterised by a vertex neighbourhood

which is a cone over a spherical Coxeter .n 1/-simplex.

The following structural result of A. Felikson and P. Tumarkin [10, Theorem A] will

be of importance later in this work. For its statement, the compact Coxeter polyhedra in

H4that are products of two simplices of dimensions greater than 1will play a certain role.

There are seven such polyhedra which were discovered by F. Esselmann [8]; see also [9]

and Examples 2,4and 10 below.

Theorem 1. Let PHnbe a compact Coxeter polyhedron. If n4and all facets of

Pare mutually intersecting, then Pis either a simplex or one of the seven Esselmann

polyhedra. If n>4, then Phas a pair of non-intersecting facets.

Fix a compact Coxeter polyhedron PHnwith its bounding hyperplanes H1; : : :; HN

as above. Denote by Gthe group generated by the reﬂections riDrHi,1iN. Then, G

is a cocompact discrete subgroup of IsomHnwith Pequal to the closure of a fundamental

domain for G. The group Gis called a (cocompact) hyperbolic Coxeter group. It follows

that Gis ﬁnitely presented with natural generating set SD ¹r1; : : : ; rNºand relations

iD1and .rirj/mij D1; (1)

where mij Dmj i 2 ¹2; 3; : : : ; 1º for i6D j. Here, mij D 1 means that the product rirj

Hyperbolic Coxeter groups and minimal growth rates in dimensions 4 and 5 729

is of inﬁnite order which ﬁts into the following geometric picture. Denote by Gr.P / D

.hei; ejin;1/2Mat.N IR/the Gram matrix of P. Then, the coefﬁcients of Gr.P / off its

diagonal can be interpreted as follows:

hei; ejin;1 D´cos 

mij if ].Hi; Hj/D

mij I

cosh lij if dH.Hi; Hj/Dlij > 0:

The matrix Gr.P / is of signature .n; 1/. Furthermore, it contains important information

about P. For example, each vertex of Pis characterised by a positive deﬁnite nn

principal submatrix of Gr.P /.

Beside the Gram matrix Gr.P /, the Vinberg graph †.P / is very useful to describe a

Coxeter polyhedron P(and its associated reﬂection group G) if the number Nof its facets

is small in comparison with the dimension n. The Vinberg graph †.P / consists of nodes

vi,1iN; which correspond to the hyperplanes Hior their unit normal vectors ei.

The number Nof nodes is called the order of †.P /. If the hyperplanes Hiand Hjare

not orthogonal, the corresponding nodes viand vjare connected by an edge with weight

mij 3if ].Hi; Hj/D

mij ; they are connected by a dotted edge (sometimes with weight

lij ) if Hiand Hjare at distance lij > 0 in Hn. The weight mij D3is omitted since it

occurs very frequently.

Since Pis compact (and hence of ﬁnite volume), the Vinberg graph †.P / is connec-

ted. Furthermore, by deleting a node together with the edges emanating from it so that

†.P / gives rise to two connected components †1and †2, at most one of the two sub-

graphs †1; †2can have a dotted edge (since otherwise, the signature condition of Gr.P /

is violated).

The subsequent examples summarise the classiﬁcation results for compact Coxeter

n-polyhedra in terms of the number NDnCk,1k3, of their facets.

Example 1. The compact hyperbolic Coxeter simplices were classiﬁed by Lannér [25]

and exist for n4, only. In the case nD4, there are precisely ﬁve simplices Liwhose

Vinberg graphs †iD†.Li/,1i5, are given in Figure 1. The simplex LDL1

described by the top left Vinberg graph (or by its Coxeter symbol Œ5; 3; 3; 3; see Sec-

tion 2.2 and [18]) will be of particular importance.

Example 2. The compact Coxeter polyhedra with nC2facets in Hnhave been classiﬁed.

The list consists of the 7 examples of Esselmann and the (gluings of) straight Coxeter

prisms due to I. Kaplinskaja; see, e.g., [9,31]. The examples of Esselmann are products

of two simplices of dimensions bigger than 1 and exist in H4, only. The prisms (and

their gluings) of Kaplinskaja exist for n5, and the list includes the Makarov prism M

based on Œ5; 3; 3; 3; 3; see Theorem B. Observe that the Vinberg graphs of all Kaplinskaja

examples (including their gluings) contain one dotted edge.

Example 3. The compact hyperbolic Coxeter polyhedra PHn,n4, with nC3

facets exist up to nD8and have been enumerated by Tumarkin [35]. For nD5, his list

comprises 16 polyhedra, and they are characterised by Vinberg graphs with exactly three

N. Bredon and R. Kellerhals 730

sssss sssss sssss

5 5 4 5 5

sss



sss

s s

A



Figure 1. The compact Coxeter simplices in H4.

s s



@



4 4

sss





Figure 2. The Vinberg graphs of an Esselmann polyhedron

EH4and of a Kaplinskaja prism KH5.





@@ 

5 5

4 4

Figure 3. The Vinberg graph of

Tumarkin’s polyhedron TH5

with one pair of disjoint facets.

(consecutive) dotted edges, up to the exceptional case TH5. The polyhedron Thas

exactly one pair of non-intersecting facets and is depicted in Figure 3.

Remark 1. By a result of Felikson and Tumarkin [11, Corollary], the Coxeter polyhedra

in Examples 1,2and 3contain all compact Coxeter polyhedra with exactly one pair of

non-intersecting facets. In particular, each compact Coxeter polyhedron PHnwith

NnC4facets has a Vinberg graph with at least two dotted edges.

Every compact Coxeter polyhedron PHngives rise to a hyperbolic Coxeter group

acting cocompactly on Hn, and each cocompact discrete group GIsomHngenerated by

ﬁnitely many hyperplane reﬂections has a fundamental domain whose closure is a compact

Coxeter polyhedron in Hn. In the sequel, we often use identical notions and descriptions

for both, the polyhedron Pand the reﬂection group G.

For further details and results about hyperbolic Coxeter polyhedra and Coxeter groups,

their geometric-combinatorial and arithmetical characterisation as well as general (non-)

existence results, we refer to the foundational work of E. Vinberg [36,37]. An overview

about the diverse partial classiﬁcation results can be found in [9].

2.2. Coxeter groups and growth rates

A hyperbolic Coxeter group GD.G; S/ with SD ¹r1; : : : ; rNºas above is the geomet-

ric realisation of an abstract Coxeter system .W; S / of rank Nconsisting of a group W

generated by a subset Sof elements s1; : : : ; sNsatisfying the relations as given by (1).

In the fundamental work [6] of Coxeter, the irreducible ﬁnite (or spherical) and afﬁne

Coxeter groups are classiﬁed. Abstract Coxeter groups are most conveniently described

Hyperbolic Coxeter groups and minimal growth rates in dimensions 4 and 5 731

by their Coxeter graphs or by their Coxeter symbols. More precisely, the Coxeter graph

†D†.W / of a Coxeter system .W; S/ has nodes v1; : : : ; vNcorresponding to the gener-

ators s1; : : : ; sNof W, and two nodes viand vjare joined by an edge with weight mij 3.

In particular, there will be no edge if mij D2and there will be an edge decorated by 1

if the product element sisjis of inﬁnite order mij D 1.

In this way, the Vinberg graph of a hyperbolic Coxeter group is a reﬁned version

of its Coxeter graph. In this context, observe that the Coxeter graph –

–—–describes the

afﬁne group z

A1and – simultaneously – is underlying the Vinberg graph  of a com-

pact hyperbolic Coxeter 1-simplex as given by any geodesic segment. Furthermore, the

reﬂection group in Isom H2associated to the compact Lambert quadrilateral with Vinberg

graph ––– is given by the Coxeter graph –

–—–––––

–—–while the Vinberg graph –

–—––––

(coinciding with its Coxeter graph) describes a non-compact hyperbolic triangle of area 

In the case that the rank Nof the Coxeter system .W; S / is small, a description by the

Coxeter symbol is more convenient. For example, Œp1; : : : ; pkwith integer labels pi3

is associated to a linear Coxeter graph with kC1edges marked by the respective weights.

The Coxeter symbol Œ.p; q; r/ describes a cyclic Coxeter graph with 3 edges of weights

p,qand r. We assemble the different symbols into a single one in order to describe the

different nature of parts of the Coxeter graph in question; see, e.g., [18, Appendix].

Example 4. The Coxeter symbols of the seven Esselmann polyhedra in H4are charac-

terised by the fact that they contain two disjoint Coxeter symbols associated to compact

hyperbolic triangles and called triangular components that are separated by at least one

edge of (ﬁnite) weight m3. Accordingly, the Esselmann polyhedron EH4as depicted

in Figure 2is described by the Coxeter symbol Œ.3; 4; 3/; 4; .3; 4; 3/. Notice that none of

the triangular components .p; q; r/, given by integers p; q; r 2such that 1

pC1

qC1

r< 1,

of the Coxeter symbols appearing in Esselmann’s list is equal to .2; 3; 7/.

For a Coxeter system .W; S/ with generating set SD ¹s1; : : : ; sNº, the (spherical)

growth series fS.t/ is deﬁned by

fS.t/ D1CX

k1

aktk;

where ak2Zis the number of words w2Wwith S-length k. For references of the

subsequent basic properties of fS.t/, see for example [17,21,23]. The series fS.t/ can be

computed by Steinberg’s formula

fS.t1/DX

WT<W

jWTj<1

.1/jTj

fT.t/ ;(2)

where WT,TS, is a ﬁnite Coxeter subgroup of W, and where W¿D ¹1º. By a result

of Solomon, the growth polynomials fT.t/ in (2) can be expressed by means of their

N. Bredon and R. Kellerhals 732

Group Exponents Growth polynomial fS.x/

An1; 2; : : : ; n 1; n Œ2; 3; : : : ; n; n C1

Bn1; 3; : : : ; 2n 3; 2n 1 Œ2; 4; : : : ; 2n 2; 2n

Dn1; 3; : : : ; 2n 5; 2n 3; n 1 Œ2; 4; : : : ; 2n 2; n

G.m/

21; m 1 Œ2; m

F41; 5; 7; 11 Œ2; 6; 8; 12

H31; 5; 9 Œ2; 6; 10

H41; 11; 19; 29 Œ2; 12; 20; 30

Table 1. Exponents and growth polynomials of irreducible ﬁnite Coxeter groups.

exponents m1D1; m2; : : : ; mpaccording to the formula

fT.t/ D

iD1

ŒmiC1:

Here we use the standard notation Œk D1CtC  C tk1with Œk; l DŒk Œl and

so on. By replacing the variable tby t1, the function Œk satisﬁes the property Œk.t/ D

tk1Œk.t1/.

Table 1lists all irreducible ﬁnite Coxeter groups together with their growth polyno-

mials up to the exceptional groups E6; E7and E8which are irrelevant for this work. Let

us add that the growth series of a reducible Coxeter system .W; S/ with factor groups

.W1; S1/and .W2; S2/such that SD.S1 ¹1W2º/[.¹1W1º  S2/, satisﬁes the product

formula fS.t/ DfS1.t/ fS2.t/.

By the above, in its disk of convergence, the growth series fS.t/ is a rational function

that can be expressed as the quotient of coprime monic polynomials p.t/; q.t/ 2ZŒt  of

equal degree. The growth rate WD.W;S / is deﬁned by

WDlim sup

k!1

pak;

and it coincides with the inverse of the radius of convergence Rof fS.t/. Since Wequals

the biggest real root of the denominator polynomial q.t/, it is a real algebraic integer.

Consider a cocompact hyperbolic Coxeter group GD.G; S/. Then, the rational func-

tion fS.t/ is reciprocal (resp. anti-reciprocal) for neven (resp. nodd); see, e.g., [23]. In

particular, for nD2and 4, one has fS.t1/DfS.t / for all t6D 0. Furthermore, a result

of Milnor [29] implies that the growth rate Gis strictly bigger than 1so that Gis of

exponential growth. More speciﬁcally, for nD2and 3,Gis either a quadratic unit or a

Salem number, that is, Gis a real algebraic integer ˛ > 1 whose inverse is a conjugate

of ˛, and all other conjugates lie on the unit circle; see, e.g., [24]. However, by a result

of Cannon [4,5] (see also [23, Theorem 4.1]), the growth rates of the ﬁve Lannér groups

acting on H4and shown in Figure 1are not Salem numbers anymore; they are so-called

Hyperbolic Coxeter groups and minimal growth rates in dimensions 4 and 5 733

Perron numbers, that is, real algebraic integers > 1 all of whose other conjugates are of

strictly smaller absolute value.

Example 5. The smallest known Salem number ˛L1:176281 with minimal polynomial

L.t/ Dt10 Ct9t7t6t5t4t3CtC1equals the growth rate Œ7;3 of the

cocompact Coxeter triangle group GDŒ7; 3 with Coxeter graph –

–—––––which in turn

is the smallest growth rate among all cocompact planar hyperbolic Coxeter groups; see

[16,21].

The second smallest growth rate among them is realised by the Coxeter triangle group

Œ8; 3 with Coxeter graph –

–—––––and appears as the seventh smallest known Salem number

1:23039 given by the minimal polynomial t10 t7t5t3C1; see [22].

As a consequence, the growth rate of the cocompact Lambert quadrilateral group

Qwith Vinberg graph   –––   is strictly bigger than Œ8;3. More precisely, the

growth rate of Qis the Salem number Q1:72208 with minimal polynomial t4

t3t2t1. Notice also that the Coxeter graph of Qequals –

–—–––––

–—–; see the proof of

Theorem Bin Section 3.

By applying similar techniques, it was shown in [19] (see also Floyd’s work [12]) that

the Coxeter triangle group with Vinberg graph –

–—––––has smallest growth rate among all

non-cocompact hyperbolic Coxeter groups of ﬁnite coarea in Isom H2, and as such it is

unique. The growth rate Œ1;3 1:32471 has minimal polynomial t3t1and equals

the smallest Pisot number ˛Sas shown by C. Smyth; see, e.g., [32] and [19, Section 3.2].

Recall that a Pisot number is an algebraic integer ˛ > 1 all of whose other conjugates are

of absolute value less than 1.

For later purpose, let us emphasize the above comparison result as follows:

Œ8;3 < Œ1;3:(3)

Example 6. Among the cocompact Coxeter tetrahedral groups, the smallest growth rate

is about 1:35098 with minimal polynomial t10 t9t6Ct5t4tC1; it is achieved

in a unique way by the group GDŒ3; 5; 3 with Coxeter graph ––––

–—––––; see [21].

Example 7. Consider the (arithmetic) Lannér group LDŒ5; 3; 3; 3 with Coxeter graph

–

–—––––––––––mentioned in Example 1. By means of Steinberg’s formula (2) and Table 1, the

growth function fL.t/ DfS.t / can be expressed according to

fL.t1/D1

fL.t/ D15

Œ2 C6

Œ2; 2 C3

Œ2; 3 C1

Œ2; 5

°1

Œ2; 2; 2 C4

Œ2; 2; 3 C2

Œ2; 2; 5 C2

Œ2; 3; 4 C1

Œ2; 6; 10 ±

Œ2; 2; 3; 4 C1

Œ2; 2; 3; 5 C1

Œ2; 2; 6; 10 C1

Œ2; 3; 4; 5 C1

Œ2; 12; 20; 30 :

It follows that

fL.t/ DŒ2; 12; 20; 30

q.t/ ;

N. Bredon and R. Kellerhals 734

where

q.t/ D1tt7Ct8t9Ct10 t11 Ct14 t15 Ct16 2t17 C2t18 t19

Ct20 t21 Ct22 t23 C2t24 2t25 C2t26 2t27 C2t28 t29 Ct30

t31 C2t32 2t33 C2t34 2t35 C2t36 t37 Ct38 t39 Ct40 t41

C2t42 2t43 Ct44 t45 Ct46 t49 Ct50 t51 Ct52 t53 t59 Ct60:

The denominator polynomial q.t/ of fL.t/ is palindromic and of degree 60. By means

of the software PARI/GP [30], one checks that q.t/ is irreducible and has – beside non-real

roots some of them being of absolute value one – exactly two inversive pairs ˛˙1; ˇ˙1of

real roots such that ˛ > ˇ > 1. Indeed, by the results in [4,5], ˛is not a Salem number

anymore. As a consequence, the growth rate LD˛1:19988 of the Lannér group

LDŒ5; 3; 3; 3 is not a Salem number. However, Œ5;3;3;3 is a Perron number. All these

properties can be checked by the software CoxIter developed by R. Guglielmetti [13,14].

Example 8. The Coxeter prism MH5found by Makarov is given by the Vinberg

graph –

–—–––––––––––––  

lwhere the hyperbolic distance lbetween the (unique) pair of

non-intersecting facets of Msatisﬁes

cosh lD1

2s7Cp5

21:07448:

In fact, the computation of lis easy since the determinant of the Gram matrix of M

vanishes. As in Example 7, one can exploit Steinberg’s formula (2) and Table 1in order to

establish the growth function fM.t/. The denominator polynomial of fM.t/ splits into the

factor t1and a certain irreducible palindromic polynomial q.t/. As above, the software

CoxIter allows us to identify the growth rate of the reﬂection group Œ5; 3; 3; 3; 3 associated

to M, as given by the largest zero of q.t/, with the Perron number M1:64759. Notice

that the factor t1is responsible for the vanishing of the Euler characteristic of M; see,

e.g., [21, (2.7)].

Example 9. For the Kaplinskaja prism KH5depicted in Figure 2, the denominator

polynomial of the growth function fK.t/ splits into the factor t1and an irreducible

palindromic polynomial q.t/ of degree 32. By means of CoxIter, one deduces that the

growth rate is a Perron number of value K2:08379.

In a similar way, one computes the individual growth series and related invariants and

properties of any cocompact (or coﬁnite) hyperbolic Coxeter group with given Vinberg

graph.

Growth rates satisfy an important monotonicity property on the partially ordered set

of Coxeter systems as follows. For two Coxeter systems .W; S/ and .W 0; S0/, one deﬁnes

.W; S/ .W 0; S0/if there is an injective map WS!S0such that mst m0

.s/.t/ for all

s; t 2S. If extends to an isomorphism between Wand W0, one writes .W; S / '.W 0; S 0/,

and .W; S/ < .W 0; S 0/otherwise. This partial order satisﬁes the descending chain condi-

Hyperbolic Coxeter groups and minimal growth rates in dimensions 4 and 5 735

tion since mst 2 ¹2; 3; : : : ; 1º where s6D t. In particular, any strictly decreasing sequence

of Coxeter systems is ﬁnite; see [28]. In this work, the following result of Terragni [34,

Section 3] will play an essential role.

Theorem 2. If .W; S/ .W 0; S0/, then .W;S / .W 0;S0/.

Example 10. Consider the seven Esselmann groups EiIsom H4,1i7, whose

Coxeter symbols consist of two triangular components separated by at least one edge of

weight m3; see Example 4. Each of their triangular components describes a cocompact

Coxeter group in Isom H2of the type .2; 3; 8/,.2; 3; 10/,.2; 4; 5/,.2; 5; 5/,.3; 3; 4/ or

.3; 3; 5/. By means of Theorem 2, we conclude that

Œ8;3 Ei; 1 i7: (4)

Notice. In the sequel, we will work with the Coxeter graph instead of the Vinberg graph

associated to a hyperbolic Coxeter group .W; S/. Hence, we replace each dotted edge

between two nodes sand s0by an edge with weight 1, just indicating that the product

element ss02Wis of inﬁnite order.

3. Growth minimality in dimensions four and ﬁve

In this section, we prove the following two results as announced in Section 1.

Theorem A. Among all Coxeter groups acting cocompactly on H4, the Coxeter simplex

group Œ5; 3; 3; 3 has minimal growth rate, and as such it is unique.

Theorem B. Among all Coxeter groups acting cocompactly on H5, the Coxeter prism

group based on Œ5; 3; 3; 3; 3 has minimal growth rate, and as such it is unique.

Proof of Theorem A.Consider a group GIsomH4generated by the set Sof reﬂections

r1; : : : ; rNin the Nfacet hyperplanes bounding a compact Coxeter polyhedron PH4.

The group GD.G; S / is a cocompact hyperbolic Coxeter group of rank N5. Assume

that the group Gis not isomorphic to the Coxeter simplex group LDŒ5; 3; 3; 3. We have

to show that G> Œ5;3;3;3 1:19988.

In view of Theorem 1, we distinguish between the two cases whether all facets of P

are mutually intersecting or not. In the case that all facets of Pare mutually intersecting,

Pis either a Lannér simplex and Gis of rank 5, or Pis one of the seven Esselmann

polyhedra with related Coxeter groups Ei,1i7, of rank 6.

(1a) The Coxeter graphs of the ﬁve Lannér simplices LDL1; : : : ; L5in H4are given

in Figure 1. The associated growth rates have been computed by means of Steinberg’s

formula and are well known; see also [4,31,33]. The software CoxIter yields the values

Œ5;3;3;4 1:38868; Œ5;3;3;5 1:51662;

Œ5;3;31;11:44970; Œ.34;4/ 1:62282;

implying that the growth rate of LDŒ5; 3; 3; 3 is strictly smaller than those of L2; : : : ; L5.

N. Bredon and R. Kellerhals 736

(1b) Let us investigate the growth rates of the Esselmann groups E1; : : : ; E7. By

Example 10, (4), we have that

Œ8;3 Ei; 1 i7:

It follows from Example 5and Example 7that

1:19988 Œ5;3;3;3 < 1:2 < Œ8;3 1:23039;

which shows that the growth rate of LDŒ5; 3; 3; 3 is strictly smaller than those of the

Esselmann groups E1; : : : ; E7.

(2) Suppose that Phas at least one pair of non-intersecting facets. Therefore, the

Coxeter graph †of Pcontains at least one edge with weight 1. Since Phas at least

N6facets, the graph †– being connected – contains a proper connected subgraph 

of order 3with weights p; q 2 ¹2; 3; : : : ; 1º of the form as depicted in Figure 4.

s s





p q

Figure 4. A subgraph of †.

By construction, the subgraph gives rise to a standard Coxeter subgroup .W; T /

of rank 3of .G; S / that satisﬁes .W; T / .G; S/. By Theorem 2, Example 5, (3), and

Example 7, we deduce in a similar way as above that

Œ5;3;3;3 < Œ8;3 < Œ1;3 †;

which ﬁnishes the proof of Theorem A.

Proof of Theorem B.Let GIsom H5be a discrete group generated by the set Sof

reﬂections r1; : : : ; rNin the Nfacet hyperplanes of a compact Coxeter polyhedron P

H5. The group GD.G; S / is a cocompact hyperbolic Coxeter group of rank N6.

Assume that Gis not isomorphic to Makarov’s rank 7prism group based on Œ5; 3; 3; 3; 3.

The associated Coxeter prism Mis described and the growth rate Mis given in Example

8. We have to show that G> M1:64759.

Inspired by the proof of Theorem A, we look for appropriate Coxeter groups of smaller

rank such that their growth data can be exploited to derive suitable lower bounds in view

of Theorem 2. To this end, consider the following abstract Coxeter groups W1; W2and

W3with generating subsets S1; S2and S3of rank 4as deﬁned by the Coxeter graphs in

Figure 5.

The Coxeter systems .Wi; Si/can be represented by hyperbolic Coxeter groups Gi

for each 1i3, and they will play an important role when comparing growth rates. In

fact, the Coxeter graph of W1coincides with the Coxeter graph of the cocompact Lambert

Hyperbolic Coxeter groups and minimal growth rates in dimensions 4 and 5 737

ssss ssss sss

1 1 1 1 1 1

Figure 5. The three abstract Coxeter groups W1; W2and W3.

quadrilateral group QIsom H2with growth rate Q1:72208; see Example 5. Since,

for the Makarov prism M, we have M1:64759, we deduce the following important

fact:

M< QDG1:(5)

Each of the remaining Coxeter groups W2and W3can be represented as a discrete sub-

group of OC.3; 1/ generated by reﬂections in the facets of a Coxeter tetrahedron of inﬁnite

volume. Indeed, one easily checks that the associated Tits form is of signature .3; 1/ and

that some of the simplex vertices are not hyperbolic but ultra-ideal points (of positive

Lorentzian norm). More importantly, the following result holds.

Lemma 1. (1) G1< G2.(2) G1< G3.

Proof. By means of Steinberg’s formula (2), we identify for each Githe ﬁnite Coxeter

subgroups with their growth polynomials according to Table 1in order to deduce the

following expressions for their growth functions fi.t/,1i3:

f1.t1/Dh.t/; (a)

f2.t1/Dh.t/ 1

Œ2; 2; 3 ;(b)

f3.t1/Dh.t/ 1

Œ2; 2; 2 :(c)

Here, the help function h.t/,t6D 0, is given by

h.t/ D14

Œ2 C3

Œ2; 2 C1

Œ2; 3 :(6)

By taking the differences between (a) and (b), (c), respectively, one obtains, for all t > 0,

f1.t1/1

f2.t1/D1

Œ2; 2; 3 > 0; 1

f1.t1/1

f3.t1/D1

Œ2; 2; 2 > 0:

For xDt12.0; 1/, we deduce that the smallest zero of 1=f1.x/ as given by the radius

of convergence of the growth series f1.x/ of G1is strictly bigger than the one of 1=f2.x/

and of 1=f3.x/. Hence, we get G1< G2and G1< G3.

For later use, we also compare the growth rate of W1DQwith the one of the Coxeter

group W4with generating subset S4of rank 4given by the Coxeter graph according to

Figure 6. Again, the group W4can be interpreted as a discrete subgroup G4OC.3; 1/

generated by the reﬂections in the facets of a Coxeter tetrahedron of inﬁnite volume.

N. Bredon and R. Kellerhals 738

ssss

414

Figure 6. The abstract Coxeter group W4.

Lemma 2. G1< G4.

Proof. We proceed as in the proof of Lemma 1and establish the growth function f4.t/ by

means of Steinberg’s formula. We obtain the following expression:

f4.t1/D14

Œ2 C3

Œ2; 2 C2

Œ2; 4 2

Œ2; 2; 4 :(d)

By means of (a), (d) and (6), we obtain the difference function

f1.t1/1

f5.t1/D1

Œ2; 3 2

Œ2; 4 C2

Œ2; 2; 4 Dt4C1

Œ2; 3 .t2C1/ > 0; 8t > 0;

and conclude as at the end of the previous proof. 

Let us return and consider a compact Coxeter polyhedron PH5with Nfacets and

associated hyperbolic Coxeter group G. By Example 1, we know that there are no compact

Coxeter simplices anymore so that N7. Furthermore, by Theorem 1,Phas at least one

pair of non-intersecting facets. In the sequel, we discuss the cases ND7,ND8and

N9.

For ND7, we are left with the three Kaplinskaja prisms (and their gluings) as given

by the Makarov prism MDW M3based on Œ5; 3; 3; 3; 3, its closely related Coxeter prism

M4based on Œ5; 3; 3; 3; 4 as well as the Coxeter prism Kwith Vinberg graph depicted in

Figure 2and treated in Example 9. By means of the software CoxIter (or some lengthy

computation), one obtains the growth rate inequalities

1:64759 M< M4< 1:84712 < K2:08379;

which conﬁrm the assertion of Theorem Bin this case.

For ND8, we dispose of Tumarkin’s classiﬁcation list comprising all compact

Coxeter polyhedra with nC3facets. For nD5, these polyhedra have Vinberg graphs

with exactly three (consecutive) dotted edges except for the polyhedron TH5depicted

in Figure 3.

The Coxeter graph associated to Tcontains the proper subgraph –

–—––

–—–which is

associated to the Coxeter group W4studied above; see Figure 6. By means of Theorem 2,

Lemma 2and (5), we deduce that

M< G4T:

For the Coxeter graph of a polyhedron Pwith 8facets in H5that is not isometric to T, we

consider its proper order 4 subgraph –

–—––

–—–. In a similar way, by Theorem 2, Lemma 1

and (5), we obtain

M< QP:

Hyperbolic Coxeter groups and minimal growth rates in dimensions 4 and 5 739

Let N9. By Remark 1, the Vinberg graph of the polyhedron PH5with Nfacets

has at least two dotted edges. However, two dotted edges are separated by an edge in view

of the signature condition of the Gram matrix Gr.P /; see Section 2.1.

Consider the Coxeter graph †of order Nof the hyperbolic Coxeter group Gassoci-

ated to P. By the above, there is a proper connected subgraph of order 4in †, depicted

in Figure 7, with weights p; q; r; s; t 2 ¹2; 3; : : : ; 1º where at least one of them is equal

to 1.

s s







rst

p q

Figure 7. The subgraph D.p; q; r; s; t/.

In view of Figure 5, describing the three Coxeter groups G1; G2and G3, and by means

of Theorem 2, the growth rate of †, and hence of P, can be estimated from below accord-

ing to

Gi†for at least one i2 ¹1; 2; 3º:

By Lemma 1and (5), we ﬁnally obtain that

M< G1P;

as desired. This ﬁnishes the proof of Theorem B.

Acknowledgements. The authors would like to thank Yohei Komori for helpful com-

ments on an earlier version of the paper.

Funding. Naomi Bredon and Ruth Kellerhals are partially supported by the Swiss

National Science Foundation 200021–172583.

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MR 1254933

Received 25 August 2020.

Naomi Bredon

Department of Mathematics, University of Fribourg, 1700 Fribourg, Switzerland;

naomi.bredon@unifr.ch

Ruth Kellerhals

Department of Mathematics, University of Fribourg, 1700 Fribourg, Switzerland;

ruth.kellerhals@unifr.ch

Canad. Math. Bull. Vol. 66 (1), 2023, pp. 232–242

http://dx.doi.org/10.4153/S000843952200025X

Canadian Mathematical Society. is is an Open Access article, distributed under the terms of the

Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits

unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly

cited.

Hyperbolic Coxeter groups of minimal

growth rates in higher dimensions

Naomi Bredon

Abstract. e cusped hyperbolic n-orbifolds of minimal volume are well known for n≤. eir

fundamental groups are related to the Coxeter n-simplex groups n. In this work, we prove that n

has minimal growth rate among all non-cocompact Coxeter groups of nite covolume in IsomHn.In

this way, we extend previous results of Floyd for n=andofKellerhalsforn=, respectively. Our

proof is a generalization of the methods developed together with Kellerhals for the cocompact case.

1 Introduction

Let Hndenote the real hyperbolic n-space with its isometry group IsomHn.

Ahyperbolic Coxeter polyhedron P ⊂Hnis a convex polyhedron of nite volume

all of whose dihedral angles are integral submultiples of π.AssociatedtoPis the

hyperbolic Coxeter group ⊂IsomHngenerated by the reections in the bounding

hyperplanes of P.Byconstruction,is a discrete group with associated orbifold

On=Hnof nite volume.

We focus on non-compact hyperbolic Coxeter polyhedra, having at least one ideal

vertex v∞∈∂Hn. Notice that the stabilizer of the vertex v∞is isomorphic to an ane

Coxeter group. e group is called non-cocompact,anditsquotientspaceOnhas at

least one cusp.

e hyperbolic Coxeter group is the geometric realization of an abstract Coxeter

system (W,S)consisting of a group Wwith a nite generating set Stogether with the

relations s=1and(ss′)mss′=1, where mss′=ms′s∈{2,3,...,∞} for all s,s′∈Swith

s= s′.egrowth series fS(t)of W=(W,S)is given by

fS(t)=1+

k≥

aktk,

where ak∈Zis the number of elements in Wwith S-length k.egrowth rate τWof

W=(W,S)is dened as the inverse of the radius of convergence of fS(t).

We are interested in small growth rates of non-cocompact hyperbolic Coxeter

groups in IsomHnfor n≥2. For n=2, Floyd [6] showed that the Coxeter group

=[3, ∞] generated by the reections in the triangle with angles π2, π3, and 0

is the (unique) group of minimal growth rate. For n=3, Kellerhals [13]provedthat

Received by the editors November 17, 2021; revised March 25, 2022; accepted March 25, 2022.

Published online on Cambridge Core April 6, 2022.

AMS subject classication: 20F55,26A12, 22E40, 11R06.

Keywords: Coxeter group, growth rate, hyperbolic Coxeter polyhedron, ane vertex stabilizer.

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions 233

Table 1: e hyperbolic Coxeter n-simplex group n.

∞









the tetrahedral group generated by the reections in the Coxeter tetrahedron with

symbol [6, 3, 3]realizes minimal growth rate in a unique way.

Consider the hyperbolic Coxeter n-simplices and their reection groups n⊂

IsomHndepicted in Table 1. For their volumes, we refer to [12]. Observe that nis of

minimal covolume among all non-cocompact hyperbolic Coxeter n-simplex groups.

e aim of this work is to prove the following result in the context of growth rates.

eorem Let 2≤n≤9. Among all non-cocompact hyperbolic Coxeter groups of nite

covolume in Isom Hn,thegroupngiven in Table 1 has minimal growth rate, and as

such the group is unique.

Our eorem should be compared with the volume minimality results for cusped

hyperbolic n-orbifolds Onfor 2 ≤n≤9. ese results are due to Siegel [16]forn=2,

Meyerho [15]forn=3, Hild and Kellerhals [10]forn=4, and Hild [9]forn≤9.

Indeed, the fundamental group of Onis related to nin all these cases.

e work is organized as follows. In Section 2.1,wesetthebackgroundabout

hyperbolic Coxeter polyhedra and their associated reection groups. Furthermore,

we present a result of Felikson and Tumarkin about their combinatorics as given by

[5, eorem B], which will play a crucial role in our proof. In fact, we will exploit the

(non-)simplicity of the Coxeter polyhedra in a most useful way. In Section 2.2,we

discuss growth series and growth rates of Coxeter groups and introduce the notion

of extension of a Coxeter graph. We also provide some illustrating examples. e

monotonicity result of Terragni [18] for growth rates, presented in eorem 2.2,will

be another major ingredient in our proof. Finally, Section 3is devoted to the proof

of our result. We perform it in two steps by assuming that the Coxeter graph under

consideration has an ane component of type 

Aor not.

2 Hyperbolic Coxeter groups and growth rates

2.1 Coxeter polyhedra and their reﬂection groups

Let Xndenote one of the geometric n-spacesofconstantcurvature,theunitn-sphere

Sn,theEuclideann-space En, or the real hyperbolic n-space Hn.Asusual,weembed

Xnin a suitable quadratic space Yn+.IntheEuclideancase,wetaketheanemodel

and write En={x∈En+xn+=0}. In the hyperbolic case, we interpret Hnas the

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

234 N. Bredon

upper sheet of the hyperboloid in Rn+,thatis,

Hn={x∈Rn+x,xn, =−1, xn+>0},

where x,xn, =x

++x

n−x

n+is the standard Lorentzian form. Its boundary

∂Hncanbeidentiedwiththeset

∂Hn={x∈Rn+x,xn, =0,

n+



k=

x

k=1, xn+>0}.

In this picture, the isometry group of Hnis isomorphic to the group O+(n,1)of

positive Lorentzian matrices leaving the bilinear form ,n, and the upper sheet

invariant.

It is well known that each isometry of Xnis a nite composition of reections

in hyperplanes, where a hyperplane H=Hvin Xnis characterized by a normal unit

vector v∈Yn+.AssociatedtoHvare two closed half-spaces. We denote by H−

vthe

half-space in Xnwith outer normal vector v.

A (convex) n-polyhedron P =∩

i∈IH−

i⊂Xnis the non-empty intersection of a nite

number of half-spaces H−

ibounded by the hyperplanes Hi=Hvifor i∈I.Afacet of P

is of the form Fi=P∩Hifor some i∈I.Inthesequel,forXn≠Sn, we always assume

that Pis of nite volume. In the Euclidean case, this implies that Pis compact, and in

the hyperbolic case, Pistheconvexhullofnitelymanypointsv,...,vk∈Hn∪∂Hn.

If vi∈Hn,thenviis an ordinary vertex,andifvi∈∂Hn,thenviis an ideal vertex of

P,respectively.

If all dihedral angles αij =(Hi,Hj)formed by intersecting hyperplanes Hi,Hj

in the boundary of Pareeitherzerooroftheform π

mij for an integer mij ≥2,

then Pis called a Coxeter polyhedron in Xn. Observe that the Gram matrix Gr(P)=

(vi,vjYn+)i,j∈Iis a real symmetric matrix with 1’s on the diagonal and non-positive

coecients o the diagonal. In this way, the theory of Perron–Frobenius applies. For

further details and references about Coxeter polyhedra in Xn, we refer to [4,19,20].

Let P=∩

i=H−

i⊂Xnbe a Coxeter n-polyhedron. Denote by ri=rHithe reection

in the bounding hyperplane Hiof P,andletG=GPbe the group generated by

r,...,rN.ItfollowsthatGis a discrete subgroup of nite covolume in IsomXn, called

ageometric Coxeter group.

AgeometricCoxetergroupG⊂IsomXnwith generating system S={r,...,rN}is

the geometric realization of an abstract Coxeter system (W,S).Infact,wehaver

i=1

and (rirj)mij =1withmij =mji ∈{2,3,...,∞} as above. Here, mij =∞indicates

that rirjis of innite order.

For Xn=Sn,Gis a spherical Coxeter group and as such nite. For Xn=En,Gis a

Euclidean or ane Coxeter group and of innite order. By a result of Coxeter [3], the

irreducible spherical and Euclidean Coxeter groups are entirely classied. In contrast

to this fact, hyperbolic Coxeter groups are far from being classied. For a survey about

partial classication results, we refer to [4].

For the description of abstract and geometric Coxeter groups, one commonly uses

thelanguageofweightedgraphsandCoxetersymbols.Let(W,S)be an abstract

Coxeter system with generating system S={s,...,sN}and relations of the form

s

i=1andsismij

j=1withmij =mji ∈{2,3,...,∞}.eCoxeter graph of the Coxeter

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Hyperbolic Coxeter groups of minimal growth rates in higher dimensions 235

Table2: ConnectedaneCoxetergraphsoforder

n+1.



An

A∞





G



Cn  

F



system (W,S)is the non-oriented graph Σ whose nodes correspond to the generators

s,...,sN.Ifsiand sjdo not commute, their nodes ni,njare connected by an edge

with weight mij ≥3. We omit the weight mij =3sinceitoccursfrequently.e

number Nof nodes is the order of Σ. A subgraph σ⊂Σ corresponds to a special

subgroup of (W,S),thatis,asubgroupoftheform(WT,T)for a subset T⊂S.

Observe that the Coxeter graph Σ is connected if (W,S)is irreducible.

In the case of a geometric Coxeter group G=(W,S)⊂IsomXn,wecallits

Coxeter graph Σ spherical,ane,orhyperbolic,ifXn=Sn,En,orHn,respectively.

In Table 2,wereproducealltheconnectedaneCoxetergraphs,usingtheclassical

notation, with the exception of the three groups 

E,

E,

E(they will not appear in

the following).

An abstract Coxeter group with a simple presentation can conveniently be

described by its Coxeter symbol. For example, the linear Coxeter graph with edges of

successive weights k,...,kN≥3isabbreviatedbytheCoxetersymbol[k,...,kN].

e Y-shaped graph made of one edge with weight pand of two strings of kand l

edges emanating from a central vertex of valency 3 is denoted by [p,3

k,l](see [12]).

Let us specify the context and consider a Coxeter polyhedron P=∩

i=H−

i⊂Hn.

Denote by =GP⊂IsomHnits associated Coxeter group and by Σ its Coxeter graph.

Since Pisofnitevolume,thegraphΣisconnected.Furthermore,ifPis not compact,

then Phas at least one ideal vertex.

Let v∈Hnbe an ordinary vertex of P.en,itslink Lvis the intersection of Pwith

asmallsphereofcentervthat does not intersect any facet of Pnot incident to v.It

corresponds to a spherical Coxeter polyhedron of Sn−and therefore to a spherical

Coxeter subgraph σof order nin Σ.

Let v∞∈∂Hnbe an ideal vertex of P. en, its link, denoted by L∞,isgivenby

the intersection of Pwith a suciently small horosphere centered at v∞as above. e

link L∞corresponds to a Euclidean Coxeter polyhedron in En−and is related to an

ane Coxeter subgraph σ∞of order ≥nin Σ.

More precisely, if v∞is a simple ideal vertex, that is, v∞is the intersection of exactly

namong the Nbounding hyperplanes of P,theCoxetergraphσ∞is connected and of

order n.Otherwise,σ∞has nc(σ∞)≥2anecomponents,andwehavethefollowing

formula:

n−1=order(σ∞)−nc(σ∞).(1)

Recall that a polyhedron is simple if all of its vertices are simple.

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

236 N. Bredon



Figure 1: e Coxeter polyhedron P⊂H.

As in the spherical and Euclidean cases, hyperbolic Coxeter simplices in Hnare

all known, and they exist for n≤9(see[1]or[20]). A list of their Coxeter graphs,

Coxeter symbols, and volumes can be found in [12]. Among the related Coxeter n-

simplex groups, the group n,asgiveninTable 1, is of minimal covolume.

e following structural result for simple hyperbolic Coxeter polyhedra due to

Felikson and Tumarkin [5, eorem B] will be a corner stone for the proof of our

eorem.

eorem 2.1 Let n ≤9,andletP⊂Hnbe a non-compact simple Coxeter polyhedron.

If all facets of P are mutually intersecting, then P is either a simplex or isometric to the

polyhedron Pwhose Coxeter graph is depicted in Figure 1.

2.2 Growth rates and their monotonicity

Let (W,S)be a Coxeter system and denote by ak∈Zthe number of elements w∈W

with S-length k.egrowth series fS(t)of (W,S)is dened by

fS(t)=1+

k≥

aktk.

In the following, we list some properties of fS(t). For references, we refer to [11].

ere is a formula due to Steinberg expressing the growth series fS(t)of a Coxeter

system (W,S)in terms of its nite special subgroups WTfor T⊆S,

fS(t−)=

WT<W

∣WT∣<∞

(−1)∣T∣

fT(t),(2)

where W∅={1}. By a result of Solomon, the growth polynomial of each term fT(t)

in (2) can be expressed by means of its exponents {m,m,...,mp}according to the

formula

fT(t)=



i=

[mi+1],(3)

where [k]=1+t++tk−and, more generally, [k,...,kr]∶=[k][kr].Acom-

plete list of the irreducible spherical Coxeter groups together with their exponents can

be found in [14]. For example, the exponents of the Coxeter group Anwith Coxeter

graph 

are {1,2,...,n}so that

fAn(t)=[2,...,n+1].(4)

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions 237

Furthermore, the growth series of a reducible Coxeter system (W,S)with factor

groups (W,S)and (W,S)such that S=(S×{1W}) ∪ ({1W}×S)satises the

product formula

fS(t)=fS(t)⋅fS(t).

In its disk of convergence, the growth series fS(t)is a rational function, which can

be expressed as the quotient of coprime monic polynomials p(t),q(t)∈Z[t]of the

same degree. e growth rate τW=τ(W,S)is dened by the inverse of the radius of

convergence of fS(t)and can be expressed by

τW=lim sup

k→∞

ak/k.

Itistheinverseofthesmallestpositiverealpoleof fS(t)and hence an algebraic

integer.

Important for the proof of our eorem is the following result of Terragni [17]

about the growth monotonicity.

eorem 2.2 Let (W,S)and (W′,S′)be two Coxeter systems such that there is an

injective map ι ∶S→S′with mst ≤m′

ι(s)ι(t)for all s,t∈S. en, τ(W,S)≤τ(W′,S′).

For n≥2, consider a Coxeter group ⊂IsomHnof nite covolume. By results of

MilnoranddelaHarpe,weknowthatτ>1. More precisely, and as shown by Terragni

[17], τ≥τ≈1.1380, where is the Coxeter simplex group given in Table 1.

Next, we introduce another tool in the proof of our result, the (simple) extension

of a Coxeter graph.

Denition 2.1 Let Σ be an abstract Coxeter graph. A (simple) extension of Σ is a

Coxeter graph Σ′obtainedbyaddingonenodelinkedwitha(simple)edgetothe

Coxeter graph Σ.

As a direct consequence of eorem 2.2,ifWis a Coxeter group with Coxeter

graph Σ, any extension Σ′of Σ encodes a Coxeter group W′such that τW≤τW′.

Example 2.3 Consider an irreducible ane Coxeter graph of order 3 as given in

Table 2.Uptosymmetry,thegraph 

Ahas a unique extension given by the Coxeter

graph at the top le in Figure 2. is graph describes the Coxeter tetrahedron [3, 3[]]

of nite volume. e Coxeter graphs 

Cand 

Ggive rise to the remaining ve

extensions depicted in Figure 2. By a result of Kellerhals [13], these six Coxeter graphs

describe Coxeter tetrahedral groups Λ of nite covolume in IsomHwhose growth

rates satisfy τΛ≥τ.

Example 2.4 In a similar way, any extension of an irreducible ane Coxeter graph

of order 4 yields a Coxeter simplex group of nite covolume in IsomH.eyare

given in Figure 3.Noticethat=[4, 3,]is part of them.

Remark 2.5 When considering irreducible ane Coxeter graphs of order greater

than or equal to 5, the resulting extensions do not always relate to hyperbolic Coxeter

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

238 N. Bredon

   

  

Figure 2: Extensions of 

A,

C,and 

G.

 



   

Figure 3: Extensions of 

A,

B,and 

C.



Figure 4: An innite volume 5-simplex.

n-simplex groups of nite covolume. For example, among the extensions of 

F,the

graph depicted in Figure 4 describes an innite volume simplex in H.

3 Proof of the Theorem

Let 2 ≤n≤9, and consider the Coxeter simplex group n⊂IsomHnwhose Coxeter

graph is depicted in Table 1. In this section, we provide the proof of our main result

stated as follows.

eorem For any 2≤n≤9,thegroupnhas minimal growth rate among all non-

cocompact hyperbolic Coxeter groups of nite covolume in IsomHn,andassuchthe

group is unique.

For n=2andforn=3, the result has been established by Floyd [6] and Kellerhals

[13]. erefore, it suces to prove the eorem for 4 ≤n≤9.

Observe that the growth rates of all Coxeter simplex groups in IsomHnare

known. eir list can be found in [17]. In particular, one deduces the following strict

inequalities:

τ≈1.1380 <<τ≈1.2481 <τ≈1.3717,(5)

τ<τ≈1.2964.(6)

For a xed dimension n,onealsochecksthatnhas minimal growth rate among

(all the nitely many) non-cocompact Coxeter simplex groups Λ ⊂IsomHn.

As a consequence, we focus on hyperbolic Coxeter groups ⊂IsomHngenerated

by at least N≥n+2 reections in the facets of a non-compact nite volume Coxeter

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions 239

∞

Figure 5: e Coxeter group W=[∞,3,3].

∞ ∞

Figure 6: e Coxeter groups W=[3, ∞,3]and W=[∞,3

,].

polyhedron P⊂Hn.Wehavetoshowthatτn<τ,whichyieldsunicityofthegroup

nwith this property.

Suppose that the Coxeter polyhedron Pis simple. By eorem 2.1,Pis either

isometric to the polyhedron P⊂IsomHdepicted in Figure 1,orPhas a pair of

disjoint facets. For the growth rate τof the Coxeter group associated to P,oneeasily

checks with help of the soware CoxIter [7,8]thatτ<τ≈2.8383. Hence, we can

assume that Pis not isometric to P.IfPhas a pair of disjoint facets, then the Coxeter

graph Σ of Pand its associated group contains a subgraph ∞.

e property that the Coxeter graph Σ contains such a subgraph of type 

A=[∞]

allows us to conclude the proof, whether the polyhedron Pis simple or not. In the

following, we rst look at this property and analyze it more closely.

3.1 In the presence of 

A

We start by considering particular Coxeter graphs of order 4 containing 

A.eir

related growth rates will be useful when comparing with the one of .isapproach

is similar to the one developed in [2].

Let W=[∞,3,3]be the abstract Coxeter group depicted in Figure 5.Bymeans

of the soware CoxIter, one checks that

τ<τW≈1.4655.(7)

Furthermore, consider the two abstract Coxeter groups W=[3, ∞,3]and W=

[∞,3

,]given in Figure 6.

For their growth rates, we prove the following auxiliary result.

Lemma 3.1 τW<τWand τW<τW.

Proof For 0 ≤i≤2, denote by fi∶= fWithe growth series of Wiand by Riits radius

of convergence. Recall that Riis the smallest positive pole of fi,andthatτWi=

Ri.

We establish the growth functions fiaccording to Steinberg’s formula (2). ey are

given as follows:



f(t−)=1−

[]+

[,]+

[,]−

[,,]−

[,,],



f(t−)=1−

[]+

[,]+

[,]−

[,,],



f(t−)=1−

[]+

[,]+

[,]−

[,,]−

[,,].

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

240 N. Bredon

Hence, for any t>0, one has the positive dierence functions given by



f(t−)−

f(t−)=

[,,]−

[,,]=t+t

[,,,]>0,



f(t−)−

f(t−)=

[,,]−

[,,]=t

[,,,]>0.

erefore, for i=1, 2, and for u=t−∈(0, 1), the smallest positive root Rof 

f(u)

is strictly bigger than the one of 

fi(u). is nishes the proof.

As a rst consequence, combining (5), (7), and Lemma 3.1,oneobtainsthat

τn<τWi,(8)

for all 4 ≤n≤9and0≤i≤2.

Next, suppose that the Coxeter graph Σ of contains a subgraph 

A.SinceΣ

is connected of order N≥n+2≥6, the subgraph 

Ais contained in a connected

subgraph σof order 4 in Σ, which is related to a special subgroup Wof .Byeorem

2.2, one has that τWi≤τWfor some 0 ≤i≤2. By combining (8) with these ndings,

and by eorem 2.2 and Lemma 3.1,onededucesthat

τn<τW≤τW≤τ.(9)

is nishes the proof of the eorem in the presence of a subgraph 

Ain Σ.

3.2 In the absence of 

A

Suppose that the Coxeter graph Σ with N≥n+2 nodes does not contain a subgraph

of type 

A.Inparticular,byeorem2.1, the corresponding Coxeter polyhedron P⊂

IsomHnisnotsimple,anditfollowsthat5≤n≤9.

Consider a non-simple ideal vertex v∞∈P.ItslinkL∞⊂En−is described by

areducibleanesubgraphσ∞with nc=nc(∞) ≥ 2 components which satises

n−1=order(σ∞)−ncby (1). In Table 3, we list all possible realizations for σ∞by

using the following notations.

Let 

σkbe a connected ane Coxeter graph of order k≥3aslistedinTable 2,and

denote by 



σkthe Coxeter graph consisting of the components of type 

σk.

Observe that for any graph 



σkin Table 3,onehas3≤min

kk≤5, and that the case

min

kk=5appearsonlywhenn=9.

Among the dierent components of σ∞, we consider the ones of smallest order ≥3

together with their extensions.

●Assume that the graph σ∞of the vertex link L∞contains an ane component



σof order 3. By Example 2.3, we know that any extension of 

σencodes a Coxeter

tetrahedral group Λ ⊂IsomHof nite covolume. e graph Σ itself contains a

subgraph σof order 4 which in turn comprises 

σ. e Coxeter graph σcorresponds

to a special subgroup Wof , and by eorem 2.2,wededucethatτΛ≤τW.

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions 241

Table 3: Reducible ane Coxeter graphs σ∞with nc≥2components

σk

of order k≥3suchthatn=order(σ∞)−nc+1.

n5 6 7 8 9



σ

σ

σ

σ

σ

σ

σ

σ

σ

σ



σ

σ

σ

σ

σ

σ



σ

σ

σ

σ

σ

σ



σ

σ

σ



σ

σ

 

τ∆≈1.678 τ∆≈1.599 τ∆≈1.668 τ∆≈1.702

Figure 7: e Coxeter groups Δi,i=1,...,4.

Since τ≤τΛ,andinviewof(5)and(6), eorem 2.2 yields the desired inequality

τn<τ≤τΛ≤τW≤τ,(10)

which nishes the proof in this case, and for n=5andn=6; see Table 3.

●Assume that the graph σ∞contains an ane component 

σof order 4. We

apply the same reasoning as above. By Example 2.4, any extension of 

σcorresponds

to a Coxeter 4-simplex group Λ of nite covolume, and τ≤τΛ.Again,Σcontains

asubgraphσcomprising 

σ. Hence, there exists a special subgroup Wof described

by σso that

τn<τ≤τΛ≤τW≤τ.(11)

By (10)and(11),theproofisnishedinthiscase,andforn=7andn=8; see

Table 3.

●Assume that σ∞contains an ane component 

σof order 5. By Table 3,one

has 7 ≤n≤9. It is not dicult to list all possible extensions of 

σ. ere are exactly

15 such extensions. It turns out that there are 11 extensions that encode Coxeter

5-simplex groups of nite covolume, whereas the remaining 4 extensions describe

5-simplex groups Δi,i=1,...,4,ofinnite covolume. ese last four simplices arise

by extending 

B,

C,and

F.eyaregiveninFigure 7, together with their associated

growth rates computed with CoxIter.

In view of (5), it turns out that

τ<τ∆i,fori=1,...,4.(12)

As above, the component 

σlies in a subgraph σof order 6 in Σ, and the latter

corresponds to a special subgroup Wof so that

either τΛ≤τWor τ∆i≤τW,1≤i≤4

https://doi.org/10.4153/S000843952200025X Published online by Cambridge University Press

242 N. Bredon

where Λ is a Coxeter 5-simplex group of nite covolume. Since τ≤τΛ,andby(5)

and (12), one deduces that

τn<τ≤τW≤τ.(13)

is nishes the proof of this case.

Finally, all the above considerations allow us to conclude the proof of the eorem.

Acknowledgment e author would like to express her gratitude to her supervisor

Ruth Kellerhals for all the expert advice and support throughout this project.

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Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland

e-mail:naomi.bredon@unifr.ch

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154

Naomi Bredon Birth: 11.10.1996

Citizenship: French

Gender: Female

Current Position

Since 12.19 Graduate Assistant,University of Fribourg.

Education

Since 12.19 PhD Studies in Mathematics,University of Fribourg.

Advisor: Prof. Dr. Ruth Kellerhals.

09.17 – 07.19 Master of Science in Mathematics ALGANT International,

University of Bordeaux.

03.15 – 07.17 Bachelor of Science in Mathematics,University of Bor-

deaux.

09.14 – 02.15 Preparatory classes MPSI,Lyc´ee Camille Julian, Bor-

deaux.

Publications

[3] Coxeter systems with 2-dimensional Davis complexes, growth

rates and Perron numbers, with T. Yukita. Accepted for publication in

Algebraic & Geometric Topology, 2022.

[2] Hyperbolic Coxeter groups of minimal growth rates in higher dimen-

sions, Can. Math. Bull. 66 (2023), 232–242.

[1] Hyperbolic Coxeter groups and minimal growth rates in dimensions

four and ﬁve, with R. Kellerhals. Groups Geom. Dyn. 16 (2022), 725–741.

Invited Talks

04.24 On the classiﬁcation of hyperbolic Coxeter polyhedra

Geometry Seminar, MPI for Mathematics in the Sciences, Leipzig.

03.24 Hyperbolic Coxeter polyhedra with mutually intersecting

facets S´eminaire Groupes et G´eom´etrie, University of Geneva.

01.24 Sur la classiﬁcation des groupes de Coxeter hyperboliques

S´eminaire de G´eom´etrie et Topologie, University of Marseille.

11.23 On the classiﬁcation of hyperbolic Coxeter polyhedra

Paroles aux jeunes chercheurs en g´eom´etrie et dynamique, GDR Platon,

University of Grenoble.

04.23 Coxeter polyhedra and reﬂection groups

Geometry graduate colloquium, ETH Zurich.

03.23 Growth rates of Coxeter systems and Perron numbers

SMS Doctoral day, University of Neuchˆatel.

01.23 Groupes de Coxeter hyperboliques et taux de croissance

S´eminaire virtuel francophone de groupes et g´eom´etrie.

12.22 Minimal growth rates for hyperbolic Coxeter groups

Geometry and Topology seminar, Israel Institute of Technology.

11.22 Hyperbolic Coxeter groups and their growth rates

Bern-Fribourg Graduate Seminar, University of Bern.

Departmental Talks

12.23 Hyperbolic Coxeter polyhedra with dihedral angles π

2,π

3and π

Oberseminar Geometrie, University of Fribourg.

11.22 Coxeter systems, growth rates and Perron numbers

Oberseminar Geometrie, University of Fribourg.

05.22 Hyperbolic manifolds and Coxeter groups

Oberseminar Geometrie, University of Fribourg.

05.22 Hyperbolic Coxeter groups of minimal growth rates

SMS Doctoral day, University of Fribourg.

12.20 Hyperbolic Coxeter groups and minimal growth rates – a follow

up. Oberseminar Geometrie, University of Fribourg.

11.20 Hyperbolic Coxeter groups and minimal growth rates in

IsomH4Oberseminar Geometrie, University of Fribourg.

Conference Participation

11.23 CUSO Graduate Colloquium, University of Fribourg.

11.23 Parole aux jeunes chercheuses et chercheurs du r´eseau Platon,

Institut Fourier, Grenoble.

07.23 Borel Seminar - Lattices in Negative Curvature, SwissMAP Re-

search Station, Les Diablerets.

03.23 Groups of Dynamical Origins, Automata and Spectra,

SwissMAP Research Station, Les Diablerets.

11.22 Discrete and locally compact groups, ENS Lyon.

07.22 Complex hyperbolic Geometry, CIRM, Luminy.

07.21 Swiss Knots, University of Fribourg.

Attended Weekly Seminar

Oberseminar Geometrie (UniFr).

Bern-Fribourg Graduate Seminar (UniFr, UniBe).

Teaching Experience

Graduate Assistant at University of Fribourg for the following courses.

Linear Algebra SP2020, SA2021-SP2022, SA2023–SP2024

Algebra and Geometry SA2020–SP2021, SA2022–SP2023

Discrete Mathematics From SP2020 to SP2024

Propedeutic Analysis 12.2019

Volontary tutoring at University of Bordeaux from 2016 to 2019.

Outreach

2022 Active participation for the “Math Week”, University of Fribourg.

2017-2019 Contribution to the events “Filles et Maths”, University of Bordeaux.

Language skills

French: Native. English: Full professional proﬁciency.

Personal interests

Climbing, Hiking, Music.