INSTITUTO NACIONAL DE MATEMÁTICA

PURA E APLICADA

DOCTORAL THESIS

On Arithmetic Manifolds withLarge Systole

Plinio G. P. Murillo

Rio de JaneiroSeptember, 2017

INSTITUTO NACIONAL DE MATEMÁTICA

PURA E APLICADA

On Arithmetic Manifolds withLarge Systole

Author:Plinio G. P. Murillo

Advisor:Mikhail Belolipetsky

A thesis submitted in fulfillment of the requirementsfor the degree of Doctor in Philosophy in Mathematics to the

Posgraduate Program in Mathematics at Instituto Nacional deMatemática Pura e Aplicada.

Rio de JaneiroSeptember, 2017

Perhaps we could, or maybe not, but if we do not try we could be sure that it willnot happen.

M. B.

Para mi madre, la mujer mas fuerte que he podido conocer, todo mi amor,admiración y respeto para ella.

CONTENTS

Abstract 5

Acknowledgments 7

1 Introduction 111.1 Systole of Riemannian manifolds . . . . . . . . . . . . . . . . 111.2 The idea of congruence coverings . . . . . . . . . . . . . . . . 121.3 Description of the problem . . . . . . . . . . . . . . . . . . . . 131.4 The results in this thesis . . . . . . . . . . . . . . . . . . . . . 151.5 Some comments about the results . . . . . . . . . . . . . . . . 161.6 Structure of this work . . . . . . . . . . . . . . . . . . . . . . . 17

2 Symmetric spaces 192.1 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . 192.2 Covering maps . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Action of Lie groups on manifolds . . . . . . . . . . . . . . . 222.5 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . 252.6 More about Lie groups . . . . . . . . . . . . . . . . . . . . . . 272.7 Locally symmetric spaces . . . . . . . . . . . . . . . . . . . . 272.8 Lattices in Lie groups . . . . . . . . . . . . . . . . . . . . . . . 28

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3 Arithmetic subgroups of algebraic groups 313.1 Affine algebraic varieties . . . . . . . . . . . . . . . . . . . . . 313.2 Linear algebraic groups . . . . . . . . . . . . . . . . . . . . . 333.3 Simply-connected algebraic groups . . . . . . . . . . . . . . . 343.4 Arithmetic groups . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Some elements from Number Theory . . . . . . . . . . . . . . 373.6 Restriction of scalars . . . . . . . . . . . . . . . . . . . . . . . 40

4 Arithmetic hyperbolic manifolds 424.1 Orthogonal groups . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Spin group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Definite real forms . . . . . . . . . . . . . . . . . . . . . . . . 454.4 The hyperboloid model of Hn . . . . . . . . . . . . . . . . . . 464.5 Arithmetic lattices in semisimple Lie groups . . . . . . . . . 484.6 Congruence subgroups . . . . . . . . . . . . . . . . . . . . . . 494.7 Arithmetic hyperbolic manifolds of the first type . . . . . . . 51

5 Systole of hyperbolic manifolds 545.1 An upper bound for systole of a hyperbolic surface . . . . . 545.2 Isometries of the hyperbolic plane . . . . . . . . . . . . . . . 555.3 Buser-Sarnak construction . . . . . . . . . . . . . . . . . . . . 575.4 Systole in higher dimensions . . . . . . . . . . . . . . . . . . 60

6 Systole of congruence coverings of arithmetic hyperbolic mani-folds 636.1 Length-trace inequality for SO(1, n)◦ . . . . . . . . . . . . . . 646.2 A trace estimate for congruence subgroups . . . . . . . . . . 656.3 A first lower bound for sys1(MI) . . . . . . . . . . . . . . . . 686.4 The displacement of elements in Spin(1, n) acting on Hn . . . 706.5 Lower bound for the displacement of congruence subgroups 736.6 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . 756.7 Sharpness of the lower bound . . . . . . . . . . . . . . . . . 776.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.8.1 Systolic genus of hyperbolic manifolds . . . . . . . . 796.8.2 Homological Codes . . . . . . . . . . . . . . . . . . . . 81

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7 Hilbert modular varieties 837.1 Hilbert modular varietes . . . . . . . . . . . . . . . . . . . . . 837.2 Upper bound for the systole growth of MI . . . . . . . . . . . 847.3 Distance estimate for congruence subgroups . . . . . . . . . 857.4 The index [Γ : Γ(I)] . . . . . . . . . . . . . . . . . . . . . . . . 907.5 The lower bound for sys1(MI) . . . . . . . . . . . . . . . . . . 917.6 Sharpness of the lower bound . . . . . . . . . . . . . . . . . . 92

Final considerations 93

References 96

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ABSTRACT

In this thesis we study the relation between the systole and the volume fortwo classes of arithmetic manifolds, namely arithmetic hyperbolic mani-folds and Hilbert modular varieties.

For any arithmetic hyperbolic n-manifold M of the first type, we provethat the systole of principal congruence coverings MI associated to primeideals I satisfy

sys1(MI) ≥8

n(n+ 1)log(vol(MI))− d1,

where d1 is a constant independent of I . This result generalizes the previ-ous results obtained in dimensions 2 and 3 in the work of Buser and Sarnakand Katz, Schaps and Vishne. As applications, we obtain explicit estimatesfor systolic genus of hyperbolic manifolds studied by Belolipetsky and thedistance of homological codes constructed by Guth and Lubotzky.

For a Hilbert modular variety Mk we prove that any principal congruencecovering MI satisfies

sys1(MI) ≥4

3√n

log(vol(MI))− d2,

for some constant d2 independent of I .

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We also prove that the constants 8n(n+1)

and 43√n

involved in the inequalitiesabove are sharp.

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ACKNOWLEDGMENTS

Without a doubt, this work would not have been possible without theguidance and support of Prof. Mikhail "Misha" Belolipetsky, whom I wouldlike to thank for his encouragement throughout these years. It has been aprivilege to work with such a leading mathematician and a great person.

It also would not have been possible to develop the skills required for aPhD project without a healthy study atmosphere. In this respect I wouldlike to thank all of the staff and employees at IMPA, whose work madeour lives much easier. I made many friends during my years at IMPAwho have also contributed to this. It is impossible to mention all of thembut I want to give special thanks to Adriana Sánchez, Alejandro Simarra,Daniel Rodriguez, David Andrade, Haimer Trejos, Heber Mesa, InocencioOrtíz, Juan David Rojas, Laura Molinas, Midory Komatsudani, NicolásMartínez, Rafael Sanabria and Yulieth Alzate, who have been part of myfamily in Brazil. My academic brothers Cayo Dória and Gisele Teixeiraalso deserve special thanks for countless conversations and constant sup-port in our mutually shared task throughout our doctorate program.

I would also like to thank the students of IMPA, all gathered there from far-reaching corners of the globe, from Argentina to Iran to Senegal to CostaRica, who have on many occasions invited me to share a beer with them,or a game of football or chess, or a good conversation. These are very pos-

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itive memories that will stay with me always.

My time in Brazil was also marked by other activities that have been im-portant for my well-being, particularly in those moments of tension andfrustration that are very often experienced during the mathematical ad-venture. I would like to thank to the family of the Batería Ritmo Avassaladorfrom the Acadêmicos da Rocinha Samba School, and the members of Salsabands Mano a Mano, Mazavana, Los Inmigrantes and La Original de MúsicaLatina for opening their arms and allowing me to share with them anotherimportant part of my life: Music. It is also beneficial to practice sports, andI would like to thank Julio José (JJ Tracer) and all the members of OmnisPro Parkour for so kindly allowing me to join them. I am indebted to theprofessors and volunteer servers of the meditation courses I attended atthe Vipassana Center Dhamma Santi. After those retreats I became muchmore calm and focused, and much better-equipped to be able to face thedifficulties that often appear during the course of academic research, andlife in general, but which can seem like huge monsters to young students.

I would like to thank professors Carlos Gustavo Moreira (Gugú), DarlanGirão, Dali Shen, Ilir Snopche and Oliver Lorscheid for accepting to be onthe Examination Committee and for having done the hard work of read-ing the full text carefully. Their many observations and corrections havehelped me to refine the precision of the text. This thesis was defendedon 23-05-2017, and as was pointed out by Prof. Illir Snopche, the date isformed by 3 prime numbers.

During the course of the research, I had the good fortune of interactingwith professors Alan Reid, Larry Guth, Matthieu Gendulphe, Ted Chin-burg, Uzi Vishne and Vincent Emery, whose interest in my work encour-aged me to continue, for which I am very grateful. I would also like tothank CIMPA and ICTP for their support, which made it possible to attendtheir research schools at Cusco-Perú, Piriápolis-Uruguay and Trieste-Italy,where some of these interactions took place and gave rise to some goodideas that appeared afterward.

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I would like to thank Coordenação de Aperfeiçoamento de Pessoal deNível Superior-CAPES for its financial support.

I would like to thank my English teacher Mariluce Pessoa for all her pa-tience and help improving my English skills.

I must give special thanks to Maria Campaña and Bonzo, who never doubtedthat this thesis would be possible.

My mother Rosario and my sister Estefanía deserve my heartfelt thanksand appreciation for their support, confidence and all their love, even froma distance.

Last but not least, I would like to thank Vanessa for all her patience, con-fidence, support and love throughout those years. My life has been muchbetter by her side.

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CHAPTER 1

INTRODUCTION

1.1 Systole of Riemannian manifolds

The study of the geometry of a spaceM means to understand the relationsbetween the geometrical elements such as length of curves, area, volume,angles, etc. In mathematics, a good place to give a precise definition ofthese concepts is to suppose that M is a Riemannian manifold.

Relations between length and area have been the object of studies thatdates back to antiquity. An example is given by the isoperimetric inequal-ity, which states that in R2 the length L of a closed curve and the area A ofthe planar region that it encloses are related by the inequality

4πA ≤ L2,

and the equality holds if and only if the curve is a circle. In recent decadesone geometrical invariant has appeared to be important in connection withdifferent domains in mathematics, it is the minimum length of closed curvesin M which cannot be continuously deformed to a point, which is calledthe systole of M and denoted by sys1(M). A natural question then is howthe systole of a Riemannian manifold M can be related to the volume ofM .

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1.2 The idea of congruence coverings

The modular group Γ = SL2(Z) is obtained by taking matrices with inte-gral entries in the group SL2(R). It is well-known that SL2(R) acts on thehyperbolic plane represented by the upper-half plane

H2 = {(x, y) ∈ R2; y > 0}

as the group of transformations which preserve the orientation and a hy-perbolic metric. The quotient S = Γ\H2 has finite area. If we fix p ∈ Z wecan consider the subgroup Γp ⊂ Γ of matrices whose reduction modulo pis the identity

Γp =

{(a b

c d

)∈ Γ | a ≡ d ≡ 1 mod p, b ≡ c ≡ 0 mod p

}.

This is a finite index subgroup of Γ which is also a normal subgroup, andit is called a (principal) congruence subgroup of Γ. The quotient Sp = Γp\H2

also has finite area and it is called a congruence covering of S. The variousproperties of the groups Γ and Γp make them appear in different branchesof mathematics as Geometry, Topology, Number Theory and Represen-tation Theory, and serve as a motivation for a general study of discretesubgroups of Lie groups.

Let M be a Riemannian manifold which is simply-connected. In mostcases Isom(M) can be represented as a subgroup G of real matrices. Withthis representation the subgroup GZ of integral matrices makes sense asthe subgroup of matrices in G with integer entries. However, this def-inition depends strongly on the choice of G and there is no reason, forexample, to expect that Λ = GZ is not just the trivial group. To avoid thisdifficulty, we must assume that G is a group satisfying polynomial equa-tions with coefficients in Q. With this assumption, a remarkable theoremby Borel and Harish-Chandra implies that the quotient M = Λ\M has fi-nite volume.

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With this well-defined notion, congruence subgroups of Λ are normal andfinite index subgroups defined by certain modular conditions, generaliz-ing those in SL2(Z) above. When Λ′ ⊂ Λ is a congruence subgroup thequotient M ′ = Λ′\M has also finite volume and it is called a congruencecovering of M .

Subgroups of G constructed as Λ and Λ′, and the quotients M and M ′ arecalled arithmetic. Among all discrete subgroups of Isom(M) the arithmeticones are of remarkable importance. Margulis proved that in many casesthe condition of Λ\M to have finite volume implies that Λ ⊂ Isom(M)

is arithmetic. We are then in a situation where the arithmetical nature ofthese spaces allows us to use number-theoretical tools in the study of theirgeometrical properties.

1.3 Description of the problem

Let S be a compact Riemann surface provided with a hyperbolic metric,which means that S is locally modelled by the hyperbolic plane H2. It iseasy to prove that S satisfies the systolic inequality (Prop. 5.1.1)

sys1(S) ≤ 2 log

(area(S)

π+ 2

). (1.1)

Using Teichmüller theory we can deform the metric on S (keeping it hy-perbolic) with the same area to obtain a new metric having arbitrary smallclosed curves, and then arbitrary small systole. So, a natural question iswhether there exist or not Riemann surfaces with systole bounded by be-low by a logarithmic function of the area.

In 1994, P. Buser and P. Sarnak used congruence subgroups to answer thisquestion. They considered certain arithmetic subgroups Γ ⊂ SL2(R) suchthat the quotients S = Γ\H2 are compact and they proved that the con-gruence coverings Sp = Γp\H2 of S associated to prime integers satisfy theinequality

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sys1(Sp) ≥4

3log(area(Sp))− c1, (1.2)

where c1 is a constant independent of p ∈ Z (Prop. 5.3.1). We note thatthe area of Sp goes to infinity with p, and so asymptotically the systole ofSp is bounded by below by 4

3log(area(Sp)). In 2013, S. Makisumi proved

that this bound is asymptotically sharp [31]. In this thesis we study thisphenomenon in higher dimensions.

In general, it is known that the systole of any hyperbolic n-manifold isbounded from above by a logarithmic function of the volume. On theother side, we do not have an analogous to Teichmüller theory in dimen-sions greater than 2. The existence of compact hyperbolic 3-manifoldswith arbitrary short closed geodesic follows from Thurston’s hyperbolicDehn surgery theorem [39, Thm. 5.8.2]. In higher dimensions, the exis-tence of compact hyperbolic manifolds with arbitrary short systole wasan open problem for a long time. In 2006, a method suggested by I. Agolsolved the problem in dimension n = 4 [1]. In 2011, M. Belolipetsky and S.Thomson adapted Agol’s suggestion and solved this problem in the gen-eral case [4]. It is worth noting that these examples are non-arithmetic. TheShort Geodesic Conjecture states that there exists a positive universal lowerbound for the systole of an arithmetic hyperbolic manifold. We refer thereader to [30, Sec. 12.3] for more details about this conjecture and its rela-tion with the Salem numbers and the Lehmer Conjecture.

In this way, it is natural to expect that congruence coverings of arithmetichyperbolic n-manifolds attain the logarithmic bound of its systole. In di-mension n = 3, M. Katz, M. Schaps and U. Vishne proved that congruencecoverings of arithmetic hyperbolic 3-manifolds satisfy

sys1(MI) ≥2

3log(vol(MI))− c2, (1.3)

where c2 is a constant independent of M [23]. In the same work they alsoproved that Inequality (1.2) remains valid for congruence coverings of anyarithmetic Riemann surface and they showed examples in which the con-

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stant c1 can be zero.

In general, it is known that there exists a constant C > 0 depending onlyon the dimension n such that the congruence coverings of an arithmetichyperbolic n-manifold satisfy

sys1(MI) ≥ C log(vol(MI))− c3, (1.4)

where c3 is a constant independent of MI (see [18, 3.C.6] and [19, Section4]). However, the proof of this inequality does not provide a value, oreven an estimate of the constant C. The question of the precise growth ofthe systole of congruence coverings of arithmetic manifolds is the centraltheme of this work.

1.4 The results in this thesis

In this thesis we prove that for any n ≥ 2 most of the arithmetic hyperbolicn-manifolds M have congruence coverings MI with

sys1(MI) ≥8

n(n+ 1)log(vol(MI))− d1, (1.5)

where d1 is a constant independent of MI . We study a particular class ofarithmetic hyperbolic manifolds, namely of the first type. This is the basicclass of such manifolds and it contains all the arithmetic hyperbolic man-ifolds in even dimensions. We remark that a second class appears in odddimensions and a third class appears only in dimension 7 (see Section 4.7).This result generalizes the result by P. Buser and P. Sarnak in dimension2 and also matches the lower bound found by M. Katz, M. Schaps and U.Vishne in dimension 3. Concerning the optimality of the lower bound in(1.5), in collaboration with Cayo Doria we proved that the constant 8

n(n+1)

cannot be improved. The precise statement of these results can be foundin Theorem A and Theorem B in Chapter 6. We also give two applicationsof Inequality (1.5). The first one gives us a more precise relation betweenthe systolic genus and the volume of congruence coverings of arithmetic

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hyperbolic manifolds that was studied by M. Belolipetsky in [3]. The sec-ond application concerns homological codes constructed by L. Guth andA. Lubotzky in [19]. We can obtain a lower bound for the distance of thesehomological codes which is quite close to the optimal one.

The theory of Arithmetic Groups is quite general and it can be used toconstruct Riemannian manifolds in a very broad way, not only with a hy-perbolic metric. In this sense, we were also interested in finding a precisegrowth of the systole of congruence coverings of Hilbert modular vari-eties, which are arithmetic manifolds locally modelled by an n-fold prod-uct (H2)n. In this case, we prove that the congruence coverings MI satisfy

sys1(MI) ≥4

3√n

log(vol(MI))− d2 (1.6)

for some constant d2 independent of MI . We also prove that the constant4

3√n

cannot be improved. These results follow from Theorem C and Theo-rem D.

1.5 Some comments about the results

In order to construct arithmetic hyperbolic n-manifolds we must find arith-metic subgroups of Isom+(Hn). One way of doing this, which may be themost natural approach, is to consider the arithmetic groups in SO(1, n)◦ 'Isom+(Hn) and take their congruence subgroups.

Following this approach we obtain congruence coverings MI of arithmetichyperbolic manifolds satisfying

sys1(MI) ≥4

n(n+ 1)log(vol(MI))− d3 (1.7)

for some constant d3 independent of MI . The precise statement can befound in Theorem A’.

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This gives us the first examples in dimensions greater than 3 with an ex-plicit value of the constant C in Inequality (1.4). However, the constant

4n(n+1)

does not match the constants 43

and 23

that were known from theprevious works by P. Buser, P. Sarnak [9] and M. Katz, M. Schapz and U.Vishne [23] in dimensions 2 and 3. It suggests the possibility of improvingthe constant 4

n(n+1)in (1.7) to 8

n(n+1). We found that this missing factor 2

can be compensated by constructing the arithmetic groups in the univer-sal covering of SO(1, n)◦, namely, the group Spin(1, n), instead of SO(1, n)◦

itself. Following this approach we obtain the manifolds satisfying Inequal-ity (1.5).

On the other hand, Inequality (1.4) holds for other locally symmetric spacesof Lie groups; however, the growth of systole in general could not be loga-rithmic on the volume. For example, the systole of a compact flat manifoldgrows at least as n

√vol(M) which is faster than a logarithmic function on

vol(M). The main reason for this difference is that, as metric spaces, a hy-perbolic manifold has an exponential growth, while a flat manifold has apolynomial growth. This suggests that, looking for non-hyperbolic arith-metic manifolds with systole growing logarithmically on the volume, itmight be better to look at locally symmetric spaces which are not hyper-bolic but which could share some properties with hyperbolic manifolds.Following this philosophy, we studied Hilbert modular varieties and weobtained the result previously stated.

1.6 Structure of this work

Since the manifolds attaining the logarithmic bound of its systole usedin this thesis are constructed as covering spaces of arithmetic manifolds,we start in Chapter 2 by reviewing basic aspects of the theory of coveringspaces. We also expose the connection between discrete subgroups of Liegroups and locally symmetric spaces. This last part also serves as a moti-vation for the theory of arithmetic groups.

Chapter 3 deals with the definition of arithmetic groups. To do so the

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necessary concepts from algebraic geometry and number theory are dis-cussed. In Chapter 4 we construct the arithmetic hyperbolic manifoldsof the first type and their congruence coverings. In Chapter 5 we studythe systole of hyperbolic manifolds, in particular, we prove the systolicinequality (1.1) for Riemann surfaces, we describe the Buser-Sarnak con-struction, and we discuss the previous results in higher dimensions.

The aim of Chapter 6 is to present the proofs of the theorems concerningsystole of congruence coverings of the aforementioned arithmetic hyper-bolic manifolds, which are stated in Theorem A’, Theorem A and TheoremB. The results concerning Hilbert modular varieties and their congruencecoverings are contained in Chapter 7, and are given in Theorem C andTheorem D.

Most of the results of this thesis appear in the articles:

1. Plinio G. P. Murillo. Systole of congruence coverings of arithmetic hy-perbolic manifolds (With an appendix by Cayo Dória and Plinio G. P.Murillo). arXiv:1610.03870.

2. Plinio G. P. Murillo. On growth of systole along congruence coveringsof Hilbert modular varieties. Algebraic & Geometric Topology, 17-5(2017), 2753–2762. DOI 10.2140/agt.2017.17.2753.

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CHAPTER 2

SYMMETRIC SPACES

This chapter starts by giving a survey of basic aspects of Riemannian man-ifolds, covering maps and the relation between subgroups of the funda-mental group with covering spaces of a differentiable manifold. Textbooksas [28], [13] and [27] contain detailed treatment of these topics. In the lastpart of the chapter, we review the definitions of symmetric and locallysymmetric spaces and their relation with discrete subgroups of Lie groups.

2.1 Differentiable Manifolds

Differentiable manifolds arise in mathematics as a generalization of thespace Rn from the point of view of calculus, where we have derivative offunctions and integration. We recall that a topological space X is called atopological manifold of dimension n (or topological n-manifold) if for any pointx ∈ X there exists an open subset U ⊂ X containing x and a homeomor-phism ϕ : U → Rn. An atlas on a topological n-manifold X is given by acollection of pairs {(Uα, ϕα)}, called charts, where the sets Uα are open setsthat cover X and

• For each index α the map ϕα : Uα → Rn is a homeomorphism.

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• For any α and β the transition maps

ϕαβ = ϕβ ◦ ϕ−1α : ϕα(Uα ∩ Uβ)→ ϕβ(Uα ∩ Uβ)

are smooth; it means that ϕαβ has continuous partial derivatives ofall orders as a map from open sets of Rn.

Any atlas on X can be extend to a maximal atlas in the sense that no morepairs satisfying the conditions above can be added. A topological mani-fold X provided with a maximal atlas is called a differentiable manifold. Inthis situation we will only say that X is an n-manifold

On a n-manifold X we can give a formal definition of tangent vectors. Ifx ∈ X and (U,ϕ) is a chart with x ∈ U the tangent space Tx(X) is definedas the equivalence classes of curves γ : (−1, 1) → X with γ(0) = x, wheretwo curves γ1 and γ2 are equivalent if the usual derivative in 0 of ϕ◦γ1 andϕ ◦ γ2 coincide. This definition does not depend on the chart ϕ, and thederivative of a curve γ at the point x is by definition its equivalence classin Tx(X). For any point x ∈ X the tangent space Tx(X) is a real vectorspace of dimension n. The analysis of differentiable manifolds is based onstudying these spaces, which provide a linear approximation (locally) ofthe manifold.

A continuous map f : X → Y between two manifolds is differentiablein x ∈ X if in local charts it is smooth as a map between open sets ofEuclidean spaces. In this case f defines a linear transformation

dfx : Tx(X)→ Tf(x)(Y )

[γ] 7→ [f ◦ γ],

which is called the derivative of f at x. The map f is smooth if it is differ-entiable in any point of X . If f has an inverse which is also differentiablethen f is called a diffeomorphism. The set of diffeomorphisms from X toitself forms a group with the composition of maps as its group law. Thisgroup is denoted by Diff(X).

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2.2 Covering maps

To study geometrical and topological properties of a manifold X , some-times it is convenient to represent X as a quotient X = Λ\X where X isthe universal covering of X and Λ is a discrete subgroup of Diff(X) iso-morphic with the fundamental group of X . This idea is formalized by thetheory of covering spaces which we start to review.

Definition 2.2.1. A covering map between two connected manifolds X andY is a smooth surjective map p : X → Y such that for any y ∈ Y thereexists an open subset V ⊂ Y containing y and a disjoint collection of opensets Uα ⊂ X such that p−1(V ) = ∪αUα and the restriction p |Uα : Uα → V isa diffeomorphism for any α.

Example 2.2.2. The map p : R → S1 given by p(t) = (cos(2πt), sin(2πt)) isa covering map.

A covering map p : X → Y can be thought as a way to parametrize locallythe manifold Y with the same local parameter p. A very useful propertyof covering maps is that of lifting of curves.

Proposition 2.2.3 (Lifting Property). Let p : X → Y be a covering map be-tween two connected manifolds X and Y . Suppose α : [0, 1] → Y any curveand x ∈ X is any point in the fiber p−1(α(0)). Then there exists a unique curveα : [0, 1]→ X such that α(0) = x and p◦ α = α. The curve α is called the liftingof α in x.

To any covering map p : X → Y we associate the group of covering trans-formations of p, which is given by Deck(p) = {f ∈ Diff(X) | p ◦ f = p}. Thisis a subgroup of Diff(X) and for any f ∈ Deck(p) the diagram

X X

Y

p

f

p

commutes. The covering map p is called normal (also called regular) if forany y ∈ Y and x, x′ ∈ p−1(y) there exist f ∈ Deck(p) such that f(x) = x′.

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2.3 Lie groups

A Lie group G is a manifold that is also a group in the algebraic sense, withthe property that the multiplication map and the inversion map are bothsmooth. If G is only a topological manifold and the group operation arecontinuous then G is called a topological group. A Lie subgroup H of a Liegroup G is a subgroup of G which is an immersed submanifold of G.

Example 2.3.1. The real number field and Euclidean space Rn with theaddition are Lie groups.

Example 2.3.2. The general linear group GLn(R) of invertible n × n ma-trices with real entries and product the multiplication of matrices is a Liegroup. Similarly the complex general linear group GLn(C) is a Lie group.

Example 2.3.3. If V is any real or complex vector space, we let GL(V ) de-note the group of invertible linear transformations from V to itself. If V isfinite-dimensional, any basis for V determines an isomorphism of GL(V )

with GL(n,R) or GL(n,C), with n = dimV , so GL(V ) is a Lie group.

If G and H are Lie groups, a Lie group homomorphism from G to H is asmooth map F : G→ H that is also a group homomorphism. It is called aLie group isomorphism if it is also a diffeomorphism, which implies that Fhas an inverse that is also a Lie group homomorphism. If G is a Lie groupand V is a vector space, any Lie group homomorphism ρ : G → GL(V )

is called a representation of G. If V is finite-dimensional, then ρ is called afinite-dimensional representation.

2.4 Action of Lie groups on manifolds

Interesting properties appear when a Lie group can be seen as a group oftransformations of a differentiable manifold. A (left) action of a Lie groupG on a manifold X is a map θ : G × X → X that satisfies θ(g1, θ(g2, x)) =

θ((g1g2, x)) and θ(e, x) = x for any g1, g1 ∈ G and x ∈ X , where e denotesthe identity element in G. A right action is defined in a similar way. Tosimplify the notation, we write θ(g, x) = g · x.

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Definition 2.4.1. Let θ be an action of a Lie group G on a manifold X .

1. The action θ is said to be smooth if θ is smooth as a map from G ×Xto X .

2. The orbit of a point x ∈ X is the set G · x = {g · x | g ∈ G} and theaction is called transitive if the orbit of any point is all of X .

3. Given x ∈ X , the isotropy group of x is the set of elements in G thatfix x.

4. The action is said to be free if the only element in G that fixes anyelement of X is the identity.

5. The action is said to be proper if the map G × X → X × X given by(g, x) 7→ (g·x, x) is proper. It means that the preimage of any compactset is compact.

An action of G on X defines an equivalence relation in the points of X ,where x and y are equivalent if there exists g ∈ G such that y = g · x. Theset of equivalence classes corresponds to the set of orbits and it is denotedby G\X .

Example 2.4.2. The natural action of GLn(R) on Rn is smooth and it hastwo orbits: {0} and Rn − {0}.

Example 2.4.3. Any representation of a Lie groupG on a finite-dimensionalvector space V is a smooth action of G on V .

Example 2.4.4. For any Lie subgroup H of a Lie group G the left multipli-cation (h, g) 7→ h · g defines an action of H on G with orbit space the rightcosets H\G. The right multiplication (g, h) 7→ g ·h−1 also defines an actionof H on G with orbit space the left cosets G/H . Both of these actions aresmooth and free. If H is closed, these actions are proper.

Actions of Lie groups on manifolds serve to construct other manifolds. Weare mainly interested in closed subgroups H of a Lie group G acting on Gby multiplication, and discrete groups acting on manifolds. In the firstcase we have the following [28, Thm 7.15].

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Theorem 2.4.5. Consider a Lie group G and let H be a closed Lie subgroup ofG. The right action of H on G is smooth, free, and proper. Therefore the left cosetspace G/H is a smooth manifold of dimension equal to dim(G) − dim(H), andthe quotient map π : G → G/H is a smooth submersion (i.e dπg is surjective forany g ∈ G).

In the case of discrete subgroups acting on manifolds we have [28, Prop.7.13].

Theorem 2.4.6. Suppose X is a connected manifold, and a discrete Lie groupΛ acts smoothly, freely, and properly on X . Then Λ\X has a unique smoothstructure such that the projection π : X → Λ\X is a covering map.

Conversely, if p : X → Y is a covering map, with the discrete topologythe group of covering transformation Deck(p) becomes a zero-dimensionalLie group which acts smoothly, freely and properly on X , and the quo-tient Deck(p)\X has a differentiable structure diffeomorphic to Y [28, Prop.7.12].

Remark. The results mentioned above follow from a more general result,which states that if a Lie group G acts smoothly, properly and freely on amanifold X , the orbit space G\X has a unique structure of a manifold andthe projection map π : X → G\X is a submersion [28, Thm. 7.15].

For any smooth manifold X , there exists a covering map p : X → X withX being a simply-connected manifold. The manifold X is the universalcovering ofX and it is unique up to a diffeomorphism. The name universalcomes from the fact that X covers any covering space of X . The groupDeck(p) is isomorphic to the fundamental group π1(X) of X and from thediscussion above we have that X = π1(X)\X . We will now summarizethe relation between covering spaces of X with subgroups of π1(X).

Theorem 2.4.7. For any subgroup H ⊂ π1(X) the natural projection

pH : H\X → X

is a covering map. The correspondence

24

{Subgroups of π1(X)

}→{

Covering maps overX}

H 7→ pH

defines a one-to-one correspondence between subgroups of π1(X) (up to conjuga-tion) and covering maps p : X ′ → X (up to diffeomorphism preserving fibers).

Moreover, the fiber p−1H (x) of any point x ∈ X has cardinality equal to the index

[π1(X) : H], and pH is a normal covering if and only if H is a normal subgroupof π1(X).

We refer the reader to [27] and [28] for more details about the theory ofcovering spaces.

2.5 Riemannian manifolds

A Riemannian metric g onX is the assignment of a positive-definite quadraticform gx to any tangent space Tx(X), so that it is compatible with the dif-ferentiable structure of X . A manifold X is called a Riemannian manifold ifX is endowed with a Riemannian metric.

Example 2.5.1. The Euclidean space Rn with the canonical inner product,and the n-sphere Sn with metric induced from Rn+1 are Riemannian man-ifolds.

Example 2.5.2. The upper-half plane H2 = {(x, y) ∈ R2; y > 0} can beendowed with the Riemannian metric ds2 = dx2+dy2

y2. This is a simply-

connected Riemmannian surface which is a model of the so-called hyper-bolic plane.

Example 2.5.3. More generally, the upper-half space

Hn = {x = (x1, . . . , xn) ∈ Rn | xn > 0}

with the metric ds2 =dx21+···+dx2n

x2nis a simply-connected Riemmannian man-

ifold which is a model of the so-called hyperbolic n-space.

25

Example 2.5.4. Let (X, g) and (Y, h) be two Riemannian manifolds. Forany (x, y) ∈ X×Y the tangent space T(x,y)(X×Y ) is canonically isomorphicto Tx(X)×Ty(Y ) and k(x,y) = gx+hy defines a Riemannian metric onX×Ycalled the product metric.

The Riemannian structure allows us to measure the length of curves on X .If α : [a, b] → X is a piecewise differentiable curve the length of α can bedefined as being

`(α) =

∫ b

a

√gα(t)(α′(t))dt

and this definition does not depend on the parametrization of α. The dis-tance between two points p, q ∈ X is the infimum of the length of piece-wise differentiable curves joining p with q. A Riemannian manifold X

is called complete if with this distance function X is complete as a metricspace.

For any smooth curve α(t) in a Riemannian manifoldX it is possible to de-fine the ”acceleration” of α as a second derivative α′′(t) extending the con-cept from the Euclidean geometry [13, Chap. III]. In this sense, a smoothcurve α(t) is called a geodesic if α′′(t) = 0 for any t. Hopf-Rinow’s The-orem [13, Chap. III. Thm. 2.8] states that in any complete Riemmanianmanifold X the geodesic curves are defined over all the real numbers andfor any two points p, q ∈ X there exist a geodesic joining these two points,whose length is equal to the distance between p and q.

A local diffeomorphism f between two Riemannian manifolds X and Y

is called a local isometry if for any point x ∈ X the corresponding differ-entiable map dfx : Tx(X) → Tf(x)(Y ) is an isometry of vector spaces. Anisometry of X is a diffeomorphism which is a local isometry. The set ofisometries from X to itself forms a group with respect to the composition,which is denoted by Isom(X).

Due to the local character of a Riemannian metric, we can transfer themetric between manifolds which are locally diffeomorphic. In particular,

26

it can be done for the universal covering p : X → X whenever Deck(p) ⊂Isom(X). In other words, if X has a Riemannian metric and π1(X) acts onX by isometries, we can induce the metric of X to a metric in the quotientΛ\X for any subgroup Λ of π1(X), in a way that the covering map π : X →Λ\X is a local isometry.

2.6 More about Lie groups

Some operations performed over a Lie groupG give again a Lie group. Forexample the connected component of G containing the identity, denotedby G◦, is a normal Lie subgroup of G [24, Thm. 2.6]. If G is connected, itsuniversal covering G has a canonical structure of a Lie group such that thecovering map p : G→ G is a Lie group homomorphism [24, Thm. 2.7]. Wewill refer to G◦ simply as the identity component of G. If H is a subgroup ofG, it is not true in general that H is a Lie group, however it holds if H isclosed [24, Thm. 2.9].

Example 2.6.1. The special linear group SLn(R) is a Lie group. In fact, itis the inverse image of 1 of the function determinant defined over GLn(R)

which is continuous.

2.7 Locally symmetric spaces

We will now introduce symmetric and locally symmetric spaces and theirrelation with discrete subgroups of Lie groups. We will follow the intro-duction to the subject by D. W. Morris [32, Chap. 1].

A connected Riemannian manifold X is said to be a symmetric space if forany x ∈ X there exists an isometry τx of X that fixes x and whose differ-ential map at x is − Id on TxX . Symmetric spaces are complete since allgeodesics can be extended over R using the maps τx. They are also ho-mogeneous, i.e. for any x, y ∈ X there exists an isometry g of X such thatg(x) = y.

27

Symmetric spaces arise as quotients of Lie groups by certain closed sub-groups. Let G be a connected Lie group, σ be an automorphism of G suchthat σ2 = Id andK be a compact subgroup ofGwhich is an open subgroupcontained in the fixed points set of σ. Then the quotient G/K is not just amanifold (Theorem 2.4.5) but it also has a structure of a symmetric space.Conversely, for any symmetric space X the isometry group G = Isom(X)

is a Lie group and X has the form G◦/K, where K is a compact subgroupof G◦.

Example 2.7.1. Let X = Rn. The semi-direct product G = SO(n) oRn actstransitively on X and the stabilizer of the origin is the compact subgroupK = SO(n). Then we can write X = G/K. In this case σ is given by themap σ : (k, v)→ (k,−v).

Example 2.7.2. For X = Sn, G = SO(n+ 1) and K = SO(n) is the stabilizerof the point en+1. Then Sn = SO(n + 1)/ SO(n). Here, we can take σ =

diag(−1,−1, . . . , 1).

Example 2.7.3. If X = H2 we can take G = SL2(R) and K = SO(2). Ingeneral, we have Hn = SO(1, n)◦/ SO(n) (see Section 4.4).

Since a product of symmetric spaces is also symmetric, we can obtain moreexamples by taking products of the symmetric spaces mentioned above.

A discrete subgroup of a Lie group is a subgroup that is a discrete set inthe subspace topology. If G/K is a symmetric space, a discrete subgroupΛ of the Lie group G which is also torsion-free acts smoothly, properlyand freely on G/K. The quotient X = Λ\G/K is said to be a locally sym-metric space. We note that this definition can be extended to any discretesubgroup of G, in which case the space Λ\G/K would contain singulari-ties, however we prefer to restrict this terminology to the torsion-free sub-groups to keep the geometrical point of view.

2.8 Lattices in Lie groups

It is now clear the importance of the study of discrete subgroups of Liegroups. From a topological point of view, discrete subgroups Λ of a Lie

28

group G such that the spaces Λ\G/K are compact are especially relevant.However, to allow a weaker assumption that Λ\G/K has a finite volumeis also interesting from a geometrical point of view. It motivates the notionof lattice.

In a Lie group, the analogous of the Lebesgue measure in Rn is the Haarmeasure [32, Prop. A3.1].

Proposition 2.8.1 (Haar Measure). Let G be a Lie group. Then there is a σ-finite Borel measure µ on G (unique up to a scalar multiple) such that

1. µ(C) is finite for any compact subset C of G.

2. µ(gA) = µ(A), for every Borel subset A ⊂ G, and every g ∈ G.

This measure is called a left Haar measure on G. Analougously, there is aunique right Haar measure with µ(Ag) = µ(A). A Lie group is said to beunimodular if the left Haar masure is also a right Haar measure. A funda-mental domain for a discrete subgroup Λ of G is a closed subset F of G suchthat

1. ΛF = G.

2. The interior F is dense in F , and its boundary F − F has Haar mea-sure zero.

3. λF ∩ F = ∅, for any non-trivial λ ∈ Λ.

The uniqueness of the (left) Haar measure µ induces a unique (up to ascalar multiple) σ-finite, G-invariant Borel measure ν on Λ\G [32, Prop.4.1.3] by putting

ν(Λ\A) := µ(A ∩ F)

A discrete subgroup Λ of G is called a lattice if ν(Λ\G) is finite. Intuitively,although Λ is discrete, it is "big enough" in the sense that the quotient Λ\Ghas finite measure.

Definition 2.8.2. Two subgroups Λ and Λ′ of a group G are called commen-surable if Λ ∩ Λ′ has finite index in Λ and Λ′.

29

Commensurability is an equivalence relation on the set of subgroups of agroup G. In a Lie group G, if Λ and Λ′ are discrete commensurable sub-groups of G then Λ is a (uniform) lattice if and only if Λ′ is a (uniform)lattice. Moreover, if Λ′ is a finite-index subgroup of Λ we obtain by com-paring the fundamental domains that

ν(Λ′\G) = [Λ : Λ′]ν(Λ\G).

In addition, in some situations the image of a lattice under a homomor-phism is also a lattice:

Proposition 2.8.3. [34, Chap. 1. Cor. 4.10] If φ : G → H is a continuousepimorphism between Lie groups with a compact kernel and Λ ⊂ G is a lattice inG, then φ(Λ) is a lattice in H .

Consider now the locally symmetric space X = Λ\G/K. The map π :

Λ\G → Λ\G/K is a fiber bundle with fiber H\K, where H = Λ ∩K, thenwe can apply the coarea formula [10, Ex. III.12(c) pag. 160] to obtain therelation

ν(Λ\G) = ν(H\K)vol(Λ\G/K).

Since the Haar meausure is finite for compact sets, we conclude that Λ is alattice in G if and only if X has finite Riemannian volume.

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CHAPTER 3

ARITHMETIC SUBGROUPS OF ALGEBRAICGROUPS

An important example of a lattice is the modular group SL2(Z) in SL2(R).The group SL2(Z) can be seen as the integer points of the Lie group SL2(R).Generalizations of this idea lead to the concept of an arithmetic lattice,which connects Number Theory with the study of symmetric spaces. Theformalization of integer points to an arbitrary Lie group falls into the groundof algebraic groups, which present the right place to define this concept.

In this chapter we survey the basic material from algebraic geometry neededto construct arithmetic lattices in Lie groups. The basic references for mostof the topics in this chapter are [37], [6] and [20]. We will follow the excel-lent exposition by V. Emery in [14] which we find very concise and clearfor a first approximation to the topic.

3.1 Affine algebraic varieties

If k is a field, we will denote by k the algebraic clousure of k. If K is an-other field containing k, then K is called a field extension of k and it willbe denoted by K|k.

31

Denote by Ank the set kn. Any polynomial f ∈ k[T1, . . . , Tn] defines a func-

tion f : Ank → k, which associates to an element (x1, . . . , xn) the value

f(x1, . . . , xn).

Definition 3.1.1. An affine algebraic variety over k, or simply an affine k-variety, is a subset X ⊂ An

k for some n of the form

X = {(x1, . . . , xn) ∈ Ank |f(x1, . . . , xn) = 0, ∀f ∈ S}

for some S ⊂ k[T1, . . . , Tn]. We say that X is defined over k. If k = k we onlysay that X is an affine algebraic variety.

The collection of affine varieties (over k) of Ank forms the closed sets of a

topology called the Zariski topology. If we consider only k-varieties in Ank

we obtain a coarser topology, called the k-Zariski topology. We call k-closed(k-open) the closed (resp. open) sets in the k-Zariski-topology. If there isno mention of the field k, the closed sets mean closed in the k-topology.

Definition 3.1.2. Let X ⊂ Ank and Y ⊂ Am

k be two affine k-varieties. Themap

φ = (φ1, . . . , φm) : X → Y

is said to be a morphism over k, or a k-morphism, if all the maps φi : X → k

are elements of k[T1, . . . , Tn]. As before, if k = k we just say a morphism.

Extension of scalars. If K|k is a field extension, we can associate to anyvariety X over k a variety defined over K by considering the same poly-nomials but replacing An

k by AnK . We say that the new variety is obtained

by extension of scalars. If the extension K|k is algebraic, the extension ofscalars of X remains the same subset of kn, but even in this situation it isimportant to keep the difference between the field of definition of X . Tospecify this we write X|k and X|K for the k-variety X and the K-varietyobtained by extension of scalars to K.

Rational points. The subset of k-rational points of the k-variety X is de-fined by

X(k) := X ∩ kn.

32

The definition of the K-rational points X(K) of X|K is similar, and wehave in particular that X = X(k). Note that, if Y is a k- variety and φ isa k-morphism between X and Y , then φ(X(k)) ⊂ Y (k). This implies thatthe rational points are preserved by an isomorphism of k-varieties.

Product variety. If X ∈ Ank and Y ∈ Am

k are affine k-varieties defined bythe sets of polynomials S1 ⊂ k[T1, . . . , Tn] and S2 ⊂ k[Tn+1, . . . , Tn+m], theproduct variety X × Y is defined by the subset S1 ∪ S2 seen as polynomialsin k[T1, . . . , Tn+m]. We have then that X × Y is an affine k-variety and it isworth noting that its Zariski topology is in general not the product topol-ogy.

3.2 Linear algebraic groups

The set of matrices of the size n × n with entries in k can be identifiedwith An2

k . The function det(Tij) is continuous in the Zariski topology andthe general linear group GLn is defined by the condition det(Tij) 6= 0. Al-though this condition does not define a closed subset in An2

k , the group GLn

becomes a closed subset of An2+1k defined by the solutions of the equation

det(Tij) · Y = 1.

With this observation GLn has a structure of k-variety. Moreover, the mapsgiven by the multiplication of matrices and inversion are k-morphisms.

Definition 3.2.1. A linear algebraic group defined over k, or simply a k-groupis an affine k-variety G with a structure of group, such that the multipli-cation and inversion are k-morphisms. A k-subgroup H of G is a subgroupH ≤ G such that H is also a k-variety.

Example 3.2.2. The special linear group SLn is a k-variety in An2

k defined bythe equation det(Tij) = 1. It is also a subgroup of GLn, and hence SLn is ak-subgroup of GLn .

33

As in the case of Lie groups, an important example of a subgroup in analgebraic k-group is given by the connected component of G, which con-tains the identity element of the group. It is a closed normal subgroup ofG called the identity component of G [37, Prop. 2.2.1] and it is denoted by G◦.

The next proposition shows that an algebraic k-group can be thought as afunctor from the fields containing k to the category of groups.

Proposition 3.2.3. Let G be an algebraic k-group. The set of rational pointsG(k) inherits a group structure from G. Moreover, for any field extension K|kthe group structure on G|k extends uniquely to a group structure on G|K.

Definition 3.2.4. A homomorphism of algebraic k-groups (or a k-homomorphism)φ : G → G′ is a k-morphism which is also a group homomorphism. A k-isomorphism is a k-morphism with an inverse which is also a k-morphism.

The adjective linear in the definition refers to the fact that the underlyingvariety G is affine, because any affine algebraic group can be embedded asa subgroup of some GLn [5, Prop. 1.10]:

Theorem 3.2.5. Let G be a k-group. Then G is k-isomorphic to a k-subgroup ofsome GLn.

3.3 Simply-connected algebraic groups

When the ground field is k = R, there is a connection between algebraicR-groups and Lie groups. For example, the first fact is that G(R) is a Liegroup with the differential structure inherited from the real numbers. Wenow state some definitions which have a counterpart in the theory of Liegroups.

Definition 3.3.1. An algebraic k-group G is called semisimple if G containsno connected, normal and solvable subgroup different from G and theidentity.

Definition 3.3.2. Let G and H be two algebraic k-groups. A k-homomorphismπ : G→ H is called a k-isogeny if π is surjective and ker(π) is finite. If k = k

we just say an isogeny.

34

Definition 3.3.3. A semisimple algebraic group G defined over a field k

of zero characteristic is called simply-connected if G is connected and anyisogeny π : H→ G is an isomorphism.

The existence of a simply-connected algebraic covering of a semisimplealgebraic group holds in zero characteristic [40, Prop. 2]:

Theorem 3.3.4. Let H be a connected and semisimple k-group defined over a fieldk of zero characteristic. Then, there exists a simply-connected algebraic k-groupG and an isogeny

π : G→ H.

Example 3.3.5. Suppose k = R. The affine k-group H = PSL2 is the quo-tient of the simply-connected k-group G = SL2 by the closed and normalsubgroup {1,−1}. The projection π : SL2 → PSL2 is an isogeny.

In general we cannot expect that if an R-group G is simply-connected thenG(R) is a simply-connected Lie group. For example, the R-group SL2

is simply-connected as an algebraic group, but the group SL2(R) is notsimply-connected as a Lie group. In general we only obtain the following.

Proposition 3.3.6. Let G be an algebraic R-group which is semisimple and simply-connected. Then G(R) is connected as a Lie group.

Sometimes it is more convenient to consider the simply-connected cover-ing of an algebraic group by the algebraic group itself. One of the mainresults in this thesis is an example of this phenomenon (see Chapter 6).

3.4 Arithmetic groups

We can now formalize the idea of integer points of an algebraic group. LetG be an algebraic group defined over the rational numbers Q. By Theorem3.2.5 we can assume that G ⊂ GLn for some n. We put

G(Z) := GLn(Z) ∩G,

where GLn(Z) denotes the invertible matrices with integer entries.

35

Definition 3.4.1. Let G be an algebraic Q-group and let K = Q or R. Asubgroup of G(K) is called an arithmetic subgroup if it is commensurablewith G(Z).

Although the definition of G(Z) depends on the embedding into GLn, dif-ferent embeddings produce subgroups that are commensurable as the nextresult shows [35, Prop. 4.1].

Proposition 3.4.2. Let ϕ : G→ G′ be an isomorphism of algebraic Q-groups. IfΓ is an arithmetic subgroup of G, then ϕ(Γ) is an arithmetic subgroup of G′.

The next fundamental theorem connects the theory of algebraic groupswith the study of locally symmetric spaces [7].

Theorem 3.4.3 (Borel, Harish-Chandra). Let G be a semisimple algebraic groupdefined over Q, and Γ be an arithmetic subgroup of G(R). Then Γ is a lattice inG(R).

Example 3.4.4. The special linear group and the symplectic group are ex-amples of algebraic groups defined over Q. Therefore

1. SLn(Z) is a lattice of SLn(R) for any n ≥ 2.

2. Sp2n(Z) is a lattice in Sp2n(R).

More sophisticated examples can be produced by using quadratic forms(cf. Chapter 4).

Example 3.4.5. Let f = 3x2 + 3y2 − z2. The orthogonal group

SOf = {C ∈ GL3 |CtJC = J, det(C) = 1}

with

J =

3 0 0

0 3 0

0 0 −1

is defined over Q because the equation CtJC = J defines polynomialswith integers coefficients. This implies that Γ = SOf (Z) is a lattice inSOf (R) ∼= Isom(H2).

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3.5 Some elements from Number Theory

One could feel tempted to define arithmetic subgroups of algebraic groupsdefined over a field k different from Q. The case of a particular interest iswhen k is a finite extension of Q. In this section we will recall some con-cepts from Number Theory that will be used in the subsequent sections.Most textbooks as [26] and [33] discuss extensively this branch of mathe-matics. We also found [38] to be a good source.

Number fields

A subfield k ⊂ C is called a number field if k has finite dimension as a vectorspace over Q. The dimension of k over Q is called the degree of the extensionand it is denoted by [k : Q].

Proposition 3.5.1. For any number field k with degree d there are exactly d

monomorphisms σ : k → C.

These monomorphisms are called the Archimedean places of k. The placeσ is real if σ(k) ⊂ R and it is complex otherwise. If σ is a complex place thenσ is also a complex place of k. Two places σ1 and σ2 are called equivalent ifσ2 = σ1. If we denote by r the number of real places and by s the numberof pairs of equivalent complex places {σ, σ}, we have that

d = r + 2s.

In this work we will mainly deal with number fields with no complexplaces. We say that a number field k is totally real if any Archimedeanplace of k is real.

Algebraic integers

An element x ∈ k is called an algebraic integer if there exists a monic polyno-mial p(t) with integer coefficients such that p(x) = 0. The set of algebraicintegers in k will be denoted by Ok.

Proposition 3.5.2. The set Ok is a ring which is a free-abelian group of rankequal to [k : Q].

37

This ring generalizes the rational integers Z. Among the properties of Okwe have that its field of fractions is equal to k, and any prime ideal ismaximal. Although the unique factorization does not hold in the elementsof Ok, this holds on its set of ideals.

Theorem 3.5.3. [38, Thm. 5.6] Every non-zero ideal I in Ok has a factorization

I = pt11 · · · ptss

where p1, . . . , ps are prime ideals and t1, . . . , ts are natural numbers. This factor-ization is unique up to the order if the ideals p1, . . . , ps are coprime.

The ideals of Ok are contained in a group where the inverses are fractionalideals. An Ok-module a is said to be a fractional ideal if there exists a non-zero c ∈ Ok such that ca is an ideal inOk. The non-zero fractional ideals ofOk form an abelian group under multiplication [38, Thm 5.5].

As in the rational integers, the factorization property allows one to reducemany questions about ideals to prime ideals. The Chinese Remainder The-orem helps in this task [26, I. §4]:

Theorem 3.5.4. Let I ⊂ Ok be an ideal and I = pt11 · · · ptss its factorization onprime ideals. Then

Ok/I ∼=s∏i=1

Ok/ptii .

Norm

Let k be a number field of degree d and σ1, . . . , σd its Archimedean placesgiven by Proposition 3.5.1. For any y ∈ k we define the norm of y by

Nk(y) :=d∏i=1

σi(y).

When the field k is clear from the context we simply write N(y). For anyelement in the ring of integers of k its norm lies in Z.

38

For any non-zero ideal I of Ok the quotient Ok/I is finite. The norm of I isdefined as

Nk(I) := |Ok/I|.

As before, we will write N(I) when the field k is clear from the context.There is no reason to confuse this norm with the norm of elements of Ok,since the latter applies only for ideals, however they coincide for principalideals [38, Cor. 5.10].

Proposition 3.5.5. If y ∈ Ok, y 6= 0, and (y) = yOk, then N((y)) = N(y).

By the uniqueness of the factorization in prime ideals and the ChineseRemainder Theorem we have that N(I · J) = N(I) N(J) for any two idealsin Ok.

Dedekind Zeta function

For any number field k, the Dedekind Zeta function is defined in the complexhalf-plane <(s) > 1 by the series

ζk(s) =∑I

1

N(I)s,

where the sum is over all the non-zero ideals I in Ok. If k = Q we recoverthe Riemann Zeta function

ζ(s) =∞∑n=1

1

ns.

The Dedekind zeta function ζk(s) can be also written as the product

ζk(s) =∏p

1

1− N(p)−s, <(s) > 1,

where the product is being taken over the prime ideals of Ok. It is calledthe product formula for ζk(s). A detailed exposition of the analytic aspectsof these functions can be found in [26, Chap VII].

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3.6 Restriction of scalars

Let k be a number field and G an algebraic k-group. By Theorem 3.2.5 wecan assume that G ⊂ GLn and we can define

G(Ok) := GLn(Ok) ∩G

where GLn(Ok) denotes the invertible matrices with entries in the ringof integers Ok. Imitating the definition for algebraic Q-groups, we couldsay that a discrete subgroup Γ of G(k) is arithmetic if Γ is commensurablewith G(Ok). It turns out that this construction does not increase the cat-egory of arithmetic groups. To see this, it is necessary to use restriction ofscalars. The idea is to construct for any algebraic group G defined overa number field k, an algebraic group Rk|Q(G) defined over Q such thatG(Ok) ∼= Rk|Q(G)(Z). We will mention the main properties of the restric-tion of scalars functor and we refer the reader to [35, 2.1.2] and [30, Sec.10.3] for the details of the construction.

Let k be a number field of degree d over Q and G be an algebraic subgroupof GLn defined over k. For simplicity we suppose that k is a totally realnumber field. If p ∈ k[x1,1, . . . , xn,n] and σ : k → R is a place of k, wedenote by pσ the polynomial in σ(k)[x1,1, . . . , xn,n] obtained by applyingσ to the coefficients of p. Since G is defined over k, there exist a subsetS ⊂ k[x1,1, . . . , xn,n] such that

G = {g ∈ GLn |p(g) = 0 for any p ∈ S}.

For any embedding σ : k → R we define the conjugate of G by

Gσ = {g ∈ GLn |pσ(g) = 0 for any p ∈ S}.

Theorem 3.6.1. [30, Sec. 10.3] Let k be a (totally real) number field of degree dover Q and let G be a semisimple algebraic k-group. The functor Rk|Q satisfies

1. Rk|Q(G) is a semisimple algebraic Q-group;

2. Up to conjugation, we have Rk|Q(G) ∼= G×Gσ2 × · · · ×Gσd ;

40

3. (Rk|Q(G))(Q) ∼= G(k) and (Rk|Q(G))(Z) ∼= G(Ok).

Example 3.6.2. Let k = Q(√

2). The group SL2 is an algebraic k-group andfor the non-trivial embedding σ : k → R sending

√2 to −

√2 we have

SLσ2 = SL2. Therefore the group SL2(Ok) of two by two matrices withentries in Ok and determinant one becomes an arithmetic subgroup of

Rk|Q(SL2)(R) ∼= SL2(R)× SL2(R).

By Theorem 3.4.3 we have that SL2(Ok) is a lattice in SL2(R) × SL2(R).The action of SL2(R) on H2 induces a natural action of SL2(R) × SL2(R)

on H2 ×H2, and the quotient SL2(Ok)\H2 ×H2 is called a Hilbert modularsurface.

Example 3.6.3. Let k = Q(√

2) again. If f = x2−√

2y2−√

2z2, in the sameway as in Example 3.4.5 the orthogonal group

SOf = {C ∈ GL3 |CtJC = J, det(C) = 1}

with

J =

1 0 0

0 −√

2 0

0 0 −√

2

is an algebraic group defined over k. For the non-trivial embedding σ :

k → R sending√

2 to −√

2 the quadratic form fσ = x2 +√

2y2 +√

2z2 andthe conjugate (SOf )

σ is equal to SOfσ . Therefore SOf (Ok) is an arithmeticsubgroup of

Rk|Q(SOf )(R) ∼= SOf (R)× SOfσ(R).

By Theorem 3.4.3 SOf (Ok) is a lattice in SOf (R) × SOfσ(R). Since fσ ispositive definite the Lie group SOfσ(R) is compact and hence Proposition2.8.3 implies that SOf (Ok) projects to a lattice in SOf (R) ∼= Isom(H2).

41

CHAPTER 4

ARITHMETIC HYPERBOLIC MANIFOLDS

In this chapter we will construct the arithmetic lattices of the first type inIsom+(Hn). These lattices will appear from integer points of orthogonalgroups and for their construction it is necessary to recall some generalaspects of quadratic forms, their associated algebraic groups, and relatethem with the geometry of the hyperbolic n-space. Some examples of theconstruction in dimension 2 were presented in Examples 3.4.5 and 3.6.3.

4.1 Orthogonal groups

Let k be a field of characteristic different from 2 and V an n-dimensionalk-vector space. A quadratic form f on V is a map f : V → k such that

1. f(λ · v) = λ2f(v) for any λ ∈ k and v ∈ V .

2. The function Φ : V × V → k given by

Φ(v, w) :=1

2(f(v + w)− f(v)− f(w))

is k-bilinear.

The map Φ is called the bilinear form associated to f . In addition, f is said tobe non-degenerate if Φ(v, w) = 0 for any w ∈ V implies v = 0. The space V

42

endowed with a quadratic form f is called a quadratic space over k.

Let (V1, f1) and (V2, f2) be two quadratic spaces. An isomorphism

T : V1 → V2

of k-vector spaces is an isometry if T also preserves the quadratic structure,in the sense that

f2(T (v)) = f1(v)

for any v ∈ V. In general, we say that two quadratic forms f and f ′ definedover k are equivalent if their corresponding quadratic spaces are isometric.

The isometries of a quadratic space V to itself form a group under com-position, the orthogonal group denoted Of (V ), or simply Of . For any basisB = {e1, . . . , en} of V , the bilinear form Φ can be represented by a matrixF with entries Fij = Φ(ei, ej). With this representation, T is an isometry of(V, f) if and only if T tFT = F , in other words

Of = {T ∈ GLn(V ) | T tFT = F}.

The equation T tFT = F determines polynomial equations with coeffi-cients in k defining the group Of . Hence, Of is an algebraic k-group. It isclear from the definition that equivalent quadratic forms have k-isomorphicorthogonal groups.

Proposition 4.1.1. [30, Lem. 0.9.4] Let (V, f) be a quadratic space over k. Then,V has an orthogonal basis such that the associated matrix F is diagonal. In otherwords, every quadratic form is equivalent to a diagonal form

d1x21 + d2x

2 + · · ·+ dnx2n.

4.2 Spin group

The identity component of the algebraic orthogonal group Of associatedto a quadratic form f is the special orthogonal group SOf of isometries withdeterminant 1. The group SOf is however not simply-connected and its

43

simply-connected algebraic covering is the spin group Spinf that we aregoing to explore in this section. We will refer the reader to [8], [11] and[12] for further details.

Clifford algebra. Let V be an n-dimensional vector space over k. Sup-pose that f : V → k is a non-degenerate quadratic form with associatedbilinear form Φ. If T (V ) denotes the tensor algebra of V and af the two-sided ideal of T (V ) generated by the elements x⊗ y+ y⊗ x− 2Φ(x, y), theClifford algebra of f is defined as the quotient C (f) = T (V )/af .

The Clifford algebra of f is a unitary associative algebra over k togetherwith a canonical map j : V → C (f) such that j(x)2 = f(x) for any x ∈ V ,which satisfies the following universal property: Given an associative k-algebra R with unity, for any map g : V → R satisfying g(x)2 = f(x) thereexists a unique k-algebra homomorphism g : C (f)→ R such that g◦j = g.

C (f)

V R

gj

g

Clifford group. We identify V with its image j(V ) in C (f) and k withk ·1 ⊂ C (f). If we choose an orthogonal basis {e1, . . . , en} of V with respectto Φ then in C (f) we have the relations e2

ν = f(eν) and eνeµ = −eµeν forµ, ν = 1, . . . , n, µ 6= ν. Let An be the set of subsets of the set {1, . . . , n} andlet Pn the subset of An given by the ordered sets M = {µ1, . . . , µk} ∈ An

with µ1 < · · · < µk. For anyM ∈Pn we write eM = eµ1 · . . . ·eµk and e∅ = 1.Every element in C (f) can be written uniquely in the form

s =∑M∈Pn

sMeM

with sM ∈ k.

The even Clifford algebra C +(f) of f is the k-subalgebra of C (f) generatedby the elements eM with |M | even. We call the Clifford group C l(f) of f

44

(resp. the special Clifford group C l+(f)) the multiplicative group of the in-vertible elements of C (f) (resp. C +(f)) such that sV s−1 = V .

The Clifford algebra C (f) has an anti-automorphism ∗ which acts on thebasis elements by (eµ1eµ2 · · · eµk)

∗ = eµkeµk−1· · · eµ1 . For any s ∈ C l+(f), ss∗

is an element of k×, which is called the spinor norm of s [8, §9, Proposition4]. The spin group of f is then the group of elements in the special Cliffordgroup with the spinor norm equal to one:

Spinf = {s ∈ C +(f), sV s∗ = V and ss∗ = 1}. (4.1)

The conditions defining the group Spinf are determined by polynomials in2n variables which give Spinf the structure of an algebraic k-group. For anelement s ∈ Spinf , the map ϕs : V 7→ V given by ϕs(x) = sxs−1 preservesthe quadratic form f and defines a homomorphism of groups

ϕ : Spinf → SOf , (4.2)

s 7→ ϕs.

The kernel of ϕ is the set {1,−1} and hence ϕ is an isogeny. Moreover if fis isotropic (i.e. f(v) = 0 for some non-zero v ∈ V ) then

SOf (k)/ϕ(Spinf (k)) ' k×/k×2

[11, II.2.3, II.2.6, II.3.3 and II.3.7]. In particular, if k is a finite field we havethat |Spinf (k)| = |SOf (k)|.

4.3 Definite real forms

As we saw in Proposition 4.1.1 any quadratic form f defined over a fieldk is equivalent to a diagonal form. If k = R we can strengthen this resultand conclude that any real quadratic form is equivalent to

r∑i=1

x2i −

n∑i=r+1

x2i .

45

The pair (r, n− r) is called the signature of f .

The first example of a real quadratic form is given by a positive-definiteform. It means that all the eigenvalues of the associated matrix are positiveand then the quadratic form has the shape

f :=n∑i=1

x2i .

In this case, the Lie group

O(n) := Of (R)

is compact, and it is isomorphic to the group of isometries of the roundsphere

Sn−1 = {(x1, . . . , xn) ∈ Rn | x21 + x2

2 + · · ·+ x2n = 1},

with the Riemannian metric inherited from Rn. The group of orientation-preserving isometries Isom+(Sn) is isomorphic to the connected Lie group

SO(n) := SOf (R),

and the map (4.2) defines a double covering map

Spinf (R) := Spin(n)→ SO(n).

In particular, the spin group Spin(n) is compact.

4.4 The hyperboloid model of Hn

The second important example of a real quadratic form is related to thegeometry of the hyperbolic space. Let now

f = f1,n := x21 − x2

2 − · · · − x2n+1.

The hyperboloid model of Hn is given by

Hn = {x ∈ Rn+1|f1,n(x) = 1, x1 > 0},

46

with the metric ds2 = −dx21 + · · ·+ dx2

n + dx2n+1. The hyperbolic n-space Hn

is a complete and simply-connected Riemannian manifold with constantsectional curvature equal to −1. A hyperbolic manifold M is a complete Rie-mannian manifold with constant sectional curvature equal to −1. Thesemanifolds can be written as the quotient spaces M = Γ\Hn, where Γ is adiscrete torsion-free subgroup of Isom(Hn).

The Lie groupSO(1, n) := SOf (R)

can be represented as

SO(1, n) = {C ∈ SLn+1(R) | CtJC = J}

where

J =

1 0 · · · 0

0 −1 · · · 0...

... . . . ...0 0 · · · −1

and Ct denotes the transpose of the matrix C. The linear action defines anaction of SO(1, n)◦ on Hn and SO(1, n)◦ is isomorphic to the orientation-preserving isometries Isom+(Hn). This action is transitive and the stabi-lizer of the point e1 = (1, 0, . . . , 0) is given by the compact subgroup K ofmatrices of the form

B =

1 0 0 · · · 0

0 b1,1 b1,2 · · · b1,n

......

... . . . ...0 bn,1 bn,2 · · · bn,n

,

where (bi,j) is an n × n orthogonal matrix. We can then identify Hn =

SO(1, n)◦/K, where K is a compact subgroup of SO(1, n)◦ isomorphic toSO(n).

The map ϕ in (4.2) induces a homomorphism of Lie groups

47

Spin(1, n) := Spinf (R)→ SO(1, n)

with its image equal to SO(1, n)◦ [11, Sec. 2.9]. This implies that Spin(1, n)

acts on the hyperbolic n-space Hn via the map ϕ in (4.2) and the image ofany lattice Γ in Spin(1, n) is a lattice in SO(1, n)◦.

4.5 Arithmetic lattices in semisimple Lie groups

Similar to the case of algebraic groups, we also have the concept of semisim-ple Lie groups. We say that two Lie groups G1 and G2 are isogenous ifthere is a finite normal subgroup Ni of a finite index subgroup G′i of Gi

for i = 1, 2, such that G′1/N1 is isomorphic to G′2/N2 . A Lie group G iscalled semisimple if G is isogenus to a finite product of simple Lie groups,where by simple we mean a non-abelian Lie group that has no nontrivial,connected, closed, proper, normal subgroups. For algebraic R-groups, thenotion of semisimplicity coincides with semisimplicity in Lie groups:

Proposition 4.5.1. Let G be an algebraic R-group. Then, G is semisimple if andonly if the Lie group G(R) is semisimple.

This shows in particular that the Lie groups SLn(R), SO(1, n) and Spin(1, n)

are semisimple Lie groups. In view of Proposition 2.8.3, restriction ofscalars and Borel and Harish-Chandra theorem motivate the definition ofarithmetic subgroups in any semisimple Lie group.

Definition 4.5.2. A discrete subgroup Γ of a semisimple Lie group G iscalled arithmetic if there exists a number field k, an algebraic k-group H,and an epimorphism ϕ : Rk|Q(H)(R) → G with compact kernel such thatϕ(H(Ok)) is commensurable to Γ, where H(Ok) = H∩GLn(Ok) with respectto some fixed embedding of H into GLn. We call k the field of definition of Γ.

Borel and Harish-Chandra’s Theorem implies that any arithmetic subgroupof a semisimple Lie group is a lattice and it allows us to connect arithmeticconsiderations to the world of geometry and topology. A lattice ofGwhichis an arithmetic subgroup is called an arithmetic lattice.

48

Example 4.5.3. Let k be a totally real number field of degree n over Q, andG = SL2 defined over k. Since Gσ(R) = G(R) for any σ : k → R, Theorem3.6.1 implies that SL2(Ok) embeds as an arithmetic subgroup in (SL2(R))n.

We have already seen in Examples 3.4.5 and 3.6.2 how to construct arith-metic lattices in some Lie groups. We can now formalize this technique inwider generality.

Proposition 4.5.4. Let k be a totally real number field k of degree d over Q, andG be a semisimple algebraic group defined over k. If Gσ(R) is compact for anynon-trivial embedding σ : k → R, then G(Ok) embeds as an arithmetic lattice ofG(R).

Proof. Recall that Rk|Q(G)(R) ' G(R)×Gσ2(R)×· · ·×Gσd(R). By Theorem3.6.1, the group G(Ok) embeds as an arithmetic lattice in Rk|Q(G)(R).Now,the compactness of Gσi(R) for i = 2, . . . , d implies that the projection onthe first factor

Rk|Q(G)(R)→ G(R)

has compact kernel. The image of this projection is equal to G(Ok), there-fore G(Ok) ⊂ G(R) is an arithmetic lattice.

4.6 Congruence subgroups

Congruence subgroups are a special class of finite-index subgroups of anarithmetic group which satisfy certain modular conditions. These sub-groups induce finite sheeted covering spaces, which are called congruencecoverings. Below we give the formal definitions.

Definition 4.6.1. Let Γ be an arithmetic subgroup of a semisimple Liegroup G which is commensurable with ϕ(H(Ok)) as in Definition 4.5.2.If I ⊂ Ok is a non-zero ideal ofOk, the principal congruence subgroup of Γ oflevel I is by definition the subgroup Γ(I) = Γ ∩ ϕ(H(I)), where

H(I) := ker(H(Ok)

πI−→ H(Ok/I))

and πI denotes the reduction map modulo I . A discrete subgroup Λ in G

is called a congruence subgroup if Γ(I) ⊂ Λ for some ideal I ⊂ Ok.

49

Since Ok/I is finite, Γ(I) is a finite-index subgroup of Γ for any ideal I ofOk. Moreover, it is a normal subgroup. In particular, any Γ(I) is commen-surable with Γ and hence Γ(I) is an arithmetic subgroup of G.

Geometrically, if Γ is torsion-free then M = Γ\G/K is a locally symmetricspace and the locally symmetric spaces MI = Γ(I)\G/K associated to thecongruence subgroups Γ(I) are regular finite sheeted covering manifoldsof M for any ideal I . If Γ is not torsion-free then M has quotient singu-larities and the covering MI → M should be interpreted in the orbifoldsense. The space M is called an arithmetic manifold (or arithmetic orbifold)and the spaces MI are called congruence coverings of M . The fact that forlarge enough I the varieties MI are smooth manifolds follows from Sel-berg’s Lemma [32, §4.8]. We remark that the definition of the subgroupΓ(I) depends on the embedding of the group H into GLn, but its commen-surability class does not depend on this choice.

Example 4.6.2. Let Γ = SL2(Z) be the modular group. For any p ∈ Z theprincipal congruence subgroup of level p is given by

Γp =

{(a b

c d

)∈ Γ | a ≡ d ≡ 1 mod p, b ≡ c ≡ 0 mod p

}Example 4.6.3. Let k = Q(

√2) and consider the quadratic form f = x2 −√

2y2 −√

2z2. Putting Γ = SOf (Ok) for any ideal I ⊂ Ok the principalcongruence subgroup at level I is represented by

Γ(I) = {(cij) ∈ Γ | cii − 1 ∈ I and cij ∈ I for i 6= j}.

The quotients Γ(I)\H2 are congruence coverings of the arithmetic surfaceΓ\H2.

Example 4.6.4. Let k = Q(√

2) and Γ = SL2(Ok). For any ideal I ⊂ Ok theprincipal congruence subgroup Γ(I) is given by

Γ(I) =

{(a b

c d

)∈ Γ | a− 1, d− 1, b, and c lie in I

}.

50

The quotients Γ(I)\H2 ×H2 are congruence coverings of the Hilbert mod-ular surface Γ\H2 ×H2 (c.f Example 3.6.2).

4.7 Arithmetic hyperbolic manifolds of the firsttype

Considering different quadratic forms with signature (1, n) over the fieldof real numbers we can represent SO(1, n)◦ ' Isom+(Hn) in different ways,and this allows us to use Proposition 4.5.4 to construct arithmetic latticesin Isom+(Hn).

Definition 4.7.1. Let k be a totally real number field. We will say that aquadratic form f defined over k is admissible if f has signature (1, n) overR, and for any non-trivial embedding σ : k → R the quadratic form fσ ispositive definite.

Recall that the notation fσ denotes the quadratic form obtained from f byapplying σ to the coefficients of f .

Example 4.7.2. Let k = Q(√

2) and f = x21 −√

2x22 − · · · −

√2x2

n+1. Forthe non-trivial place σ : k → R we have fσ = x2

1 +√

2x22 + · · · +

√2x2

n+1.

Therefore f is an admissible quadratic form.

The conditions for a quadratic form f defined over k being admissibleimply that SOf is a semisimple algebraic group defined over k such thatSOf (R) ' SO(1, n) and SOfσ is compact for any non-trivial embeddingσ : k → R. By Proposition 4.5.4 we have that SOf (Ok) embeds as anarithmetic subgroup of SO(1, n) and intersecting with SO(1, n)◦ we obtainan arithmetic lattice in Isom+(Hn).

Definition 4.7.3. The subgroups Γ of Isom+(Hn) constructed in this wayand subgroups commensurable with them are called arithmetic subgroupsof Isom+(Hn) of the first type. If Γ is torsion-free, M = Γ\Hn is called anarithmetic hyperbolic manifold of the first type.

51

This construction of arithmetic subgroups of Isom+(Hn) requires to con-sider isomorphisms SOf (R) ' SO(1, n). Since f has signature (1, n) overthe real numbers, fixing some basis, the subgroups SOf (R) and SO(1, n)

are conjugate in GLn+1(R) and we fix this isomorphism. This choice isparticularly important because it is a trace-preserving map.

The advantage of using quadratic forms to construct arithmetic latticesin Isom+(Hn) is that we can also construct these lattices from the groupSpin(1, n) instead of the orthogonal group SO(1, n).

In a similar way as in SO(1, n)◦, we can apply Proposition 4.5.4 to con-struct arithmetic subgroups of Spin(1, n). Let k be a totally real numberfield and f be an admissible quadratic form defined over k as before. SinceSpinf (R) ' Spin(1, n) and Spinfσ(R) is compact for any non-trivial σ, wehave that Spinf (Ok) embeds as an arithmetic subgroup of Spin(1, n). Aswe will see in Section 6.5, via the map ϕ in (4.2) the groups Spinf (Ok) andSOf (Ok) are commensurable. This proves that Spinf (Ok) projects to anarithmetic subgroup of the first type in Isom+(Hn).

Defining these arithmetic subgroups in Spin(1, n) requires to consider iso-morphisms Spinf (R) ' Spin(1, n). We will need such an isomorphismpreserving the real part of the spin elements. For completeness, we de-scribe one of these isomorphisms.

Suppose f is an admissible quadratic form of dimension n + 1. Let (V, f)

be a quadratic vector space of dimension n+ 1 over R and (Rn+1, q) be theEuclidean space with the quadratic form q = x2

1 − x22 − · · · − x2

n+1. Choosean orthogonal basis i1, . . . , in+1 of (V, f) such that f(x) = x2

1 − x22 − · · · −

x2n+1 and let e1, . . . , en+1 be the canonical basis of Rn+1. By the universal

property of C (f), the map g : V → C (q) defined as g(Σxjij) = Σxjej

extends uniquely to an R-algebras homomorphism g : C (f)→ C (q) given

52

by

g

∑M∈Pn+1

sM iM

=∑

M∈Pn+1

sMeM .

From this equation it is clear that g is an isomorphism and g(s)R = sR.Note that g commutes with ∗ and restricts to an isomorphism of groups

g : Spinf (R)∼−→ Spin(1, n).

We remark that any non-cocompact arithmetic lattice in Isom+(Hn) is of thefirst type and defined over Q [29, Secs. 1-2], and any arithmetic subgroupof Isom+(Hn) defined over Q is non-cocompact if n ≥ 4 [32, Sec. §6.4]. If nis even, all the co-compact arithmetic lattices of Isom+(Hn) are of the firsttype. In odd dimensions n there is a second class of co-compact arithmeticsubgroups of Isom+(Hn) arising from skew-hermitian forms over divisionquaternion algebras. In dimension 7 there is a third class constructed us-ing certain Cayley algebras [29, Sec. 1].

53

CHAPTER 5

SYSTOLE OF HYPERBOLIC MANIFOLDS

In this chapter we will present some known results concerning the relationbetween the systole and volume of hyperbolic manifolds. In particular, wewill present the construction used by P. Buser and P. Sarnak to producehyperbolic surfaces with systole bounded by below by a logarithmic func-tion of the area. We also discuss the previous results in higher dimensionswhich motivated the work in this thesis.

5.1 An upper bound for systole of a hyperbolicsurface

The hyperbolic space is the model of a complete Riemannian manifoldwith constant negative curvature. A hyperbolic surface is a differentiablesurface S with a Riemmanian metric which is locally isometric to the hy-perbolic plane H2. In other words S = Λ\H2 where Λ is a discrete torsion-free subgroup of Isom+(H2) isomorphic to the fundamental group π1(S).It is well known that the genus of a hyperbolic surface is bigger than one.

We recall that the systole of a Riemannian manifold M is the least lengthof a non-contractible closed geodesic in M , and it is denoted by sys1(M).

54

Proposition 5.1.1. Let S be a compact hyperbolic surface. Then

sys1(S) ≤ 2 log

(area(S)

π+ 2

).

Proof. For any compact Riemmanian surface S the injectivity radius isequal to sys1(S)

2, i.e for any point p ∈ S, the ball centered at p of radius

sys1(S)2

is an embedded ball. Now, the area of a ball BR of radius R inH2 is equal to 2π(cosh(R) − 1). For any embedded ball BR we have thatarea(S) ≥ area(BR) and then, taking R = sys1(S)

2we get

area(S)

2π+ 1 ≥ cosh

(sys1(S)

2

)≥ e

sys1(S)2

2,

and we obtain the desired formula applying the logarithm function.

5.2 Isometries of the hyperbolic plane

We need to recall some properties of elements in the Lie group PSL2(R),which is isomorphic to Isom+(H2). This group and its discrete subgroupshave been thoroughly investigated and a detailed study of them can befound in, for example, [22], [2] and [36].

We recall that the upper-half plane model of the hyperbolic plane is the set

H2 = {z ∈ C;=(z) > 0}

equipped with the metric ds2 = dx2+dy2

y2. The group PSL2(R) acts on H2 by

isometries by the fractional linear transformations:

Bz =az + b

cz + d, if B =

(a b

c d

)and z ∈ H2.

An element B ∈ PSL2(R) is called elliptic if it has a fixed point in H2,parabolic if it has no fixed points in H2 and has only one fixed point in ∂H2,and hyperbolic if it has no fixed points in H2 and has two fixed points in∂H2. An equivalent description is the following:

55

• B is elliptic if and only if |tr(B)| < 2;

• B is parabolic if and only if |tr(B)| = 2;

• B is hyperbolic if and only if |tr(B)| > 2.

where tr(B) denotes the trace of the matrix B. The lengths of closedgeodesics in a hyperbolic surface Λ\H2 are related to the hyperbolic ele-ments in Λ. Given a hyperbolic transformation B, the translation length ofB, denoted by `B, is defined by

`B = inf{dH2(z, Bz); z ∈ H2}.

This infimum is attained at points on the unique geodesic αB in H2 joiningthe fixed points of B in ∂H2. The transformation B leaves αB invariantand acts on it as a translation.

In particular, every hyperbolic elementB ∈ Λ determines a non-contractibleclosed geodesic α in the hyperbolic surface Λ\H2, whose length is equal tothe translation length `B ofB. Reciprocally, any closed geodesic α on Λ\H2

lifts to a geodesic αB in H2 fixed by a hyperbolic element B ∈ Λ.

Proposition 5.2.1. If B is a hyperbolic element in PSL2(R), then

2 cosh

(`B2

)= |tr(B)|.

In particular, for any z ∈ H2 we get

dH2(z,Bz) ≥ 2 log(|tr(B)| − 1) > 0. (5.1)

Proof. Since B is hyperbolic, the conditions |tr(B)| > 2 and det(B) = 1

imply that the eigenvalues λ and λ−1 of B are two different real numbers,and then B is conjugate in PSL2(R) to the matrix

C =

(λ 0

0 λ−1

).

56

Since conjugation is an isometry the translation lengths of B and C areequal. Now, C preserves the geodesic joining i and iλ2 and, since the hy-perbolic distance between i and iλ2 is equal to 2 log(λ), we obtain that|λ| = e

`B2 .

5.3 Buser-Sarnak construction

We now present the method used by Peter Buser and Peter Sarnak to con-struct hyperbolic surfaces with systole bounded from below by a logarith-mic function of the area [9, Sec. 4].

Let a and b be two square-free positive integers and considerA, the algebraover Q generated by 1, i, j, k, with the relations

i2 = a, j2 = b, ij = −ji = k.

The algebra A is a quaternion algebra and any element in X ∈ A can bewritten uniquely in the form X = X0 + X1i + X2j + X3k. Choose a and b

such that the form

N(X) = X20 − aX2

1 − bX22 + abX2

3

does not represent zero for X = (X0, X1, X2, X3) ∈ Q4, X 6= 0 (e.g a = 2,b = 5). In this case A is a division algebra (i.e. any non-zero element in Ahas an inverse in A). Let

A(Z) = {X ∈ A | Xi ∈ Z for i = 0, 1, 2, 3}

and denote by Γ the subgroup ofA(Z) withN(X) = 1. For p an odd prime,define the subgroup Γ(p) of Γ by

Γ(p) = {X ∈ Γ | X0 ≡ 1 mod p,X1 ≡ X2 ≡ X3 ≡ 0 mod p}.

The groups Γ and Γ(p) can be embedded into SL2(R) via the map

57

X0 +X1i+X2j +X3k 7→

(X0 +X1

√a X2 +X3

√a

b(X2 −X3

√a) X0 −X1

√a

).

The corresponding subgroups of PSL2(R) are denoted by Γ and Γ(p). Inthis way, Γ is an arithmetic subgroup of PSL2(R) defined over Q, whichis co-compact because A is a division algebra [22, Thm. 5.4.1], and thesubgroup Γ(p) is the congruence subgroup of Γ at level p.

Proposition 5.3.1. [9, Sec. 4] The quotients Sp = Γ(p)\H2 are hyperbolic sur-faces and they satisfy

sys1(Sp) ≥4

3log(area(Sp))− c,

for some constant c independent of p.

Proof. We saw in Proposition 5.2.1 that the trace of the elements in Γ(p)

bounds the length of closed geodesics in Sp. If

α = X0 +X1i+X2j +X3k ∈ Γ(p),

then p divides X1, X2, X3 and also

1 = N(α) = X20 − aX2

1 − bX22 + abX2

3 .

Therefore X20 ≡ 1 mod p2, and so we obtain that X0 ≡ ±1 mod p2. Now,

since Γ is a cocompact lattice in PSL2(R) it has no parabolic elements [22,Cor. 4.2.7]. Hence,

|X0| ≥ p2 − 1 (5.2)

if α 6= ±1. Equivalently, for any γ ∈ Γ(p) we have that

|tr(γ)| ≥ 2p2 − 2.

In particular, for any odd p any element γ ∈ Γ(p) is hyperbolic and so thequotient Sp = Γ(p)\H2 is a compact hyperbolic surface. Since p ≥ 3 wehave 2p2 − 3 ≥ p2 and Proposition 5.2.1 implies

sys1(Sp) ≥ 4 log(p). (5.3)

58

By the Gauss-Bonnet theorem area(Sp) = 2π(2g(Sp) − 2) where g(Sp) de-notes the genus of Sp,which can be computed using the Riemann-Hurwitzformula

g(Sp) =1

2p(p− 1)(p+ 1) · g(Γ\H2) + 1

= p(p− 1)(p+ 1) · ν + 1

where ν > 0 depends only on a and b. Therefore

p3 ≥ g(Sp)− 1

ν.

Substituing in Inequality (5.3) and applying the Gauss-Bonnet Theoremwe have that

sys1(Sp) ≥4

3log(area(Sp))− c,

with c = log(4πν).

Considering quaternion division algebras defined over totally real numberfields we obtain all the co-compact arithmetic subgroups of PSL2(R) [22,Chap. 5]. We can try to prove the same result in this general case, howeverthe difficulty in imitating Buser-Sarnak’s construction for the general caseis that the ring of integers of the number field is dense in R. We observethat the discreteness of Z in R was crucial for bounding below the traceof the elements in Γ(p) (Equation (5.2)). In 2007, M. Katz, M. Schaps andU. Vishne found a new way to bound the trace in the general setting andthey extended Proposition 5.3.1 to any co-compact arithmetic subgroup ofPSL2(R).

Theorem 5.3.2. [23, Thm. 1.5] Let Γ be an arithmetic co-compact subgroupof PSL2(R). Then for a suitable constant c depending only on Γ, the principalcongruence subgroups of Γ satisfy

sys1(XI) ≥4

3log(area(XI))− c

where XI = Γ(I)\H2 is the associated hyperbolic surface.

59

These results show that congruence coverings of a compact arithmetic hy-perbolic surface with sufficiently large area satisfy the systolic bounds

4

3log(area(S))− c1 ≤ sys1(S) ≤ 2 log(area(S)) + c2

for some constants c1, c2 which do not depend on S. In the same article[23] the authors proved that a special class of hyperbolic surfaces, the socalled Hurwitz surfaces, satisfy

sys1(S) ≥ 4

3log(area(S))

when the area of S is suficientely large [23, Thm. 1.10]. However, it is notknown whether there are examples of a Riemannian surface S with sys-tole bigger than λ log(area(S)) for some constant λ ∈ (4

3, 2). S. Makisumi

showed in 2012 that λ = 43

is optimal for congruence coverings. The mainresult in [31] can be stated in the following way.

Theorem 5.3.3. [31, Thm. 1.6] Let Γ be an arithmetic co-compact subgroup ofPSL2(R). Then, any sequence congruence subgroups of Γ eventually satisfy

sys1(XI) ≤4

3log(area(XI)) + d,

where XI = Γ(I)\H2 is the associated hyperbolic hyperbolic surface and d is aconstant that does not depend on XI .

5.4 Systole in higher dimensions

The volume of an n-dimensional hyperbolic ball Bnr of radius r > 0 is givenby the formula [36, Ex. 3.4.6, page 79]

vol(Bnr ) = vol(Sn−1)

∫ r

0

sinhn−1(t)dt,

where vol(Sn−1) denotes the volume of the Euclidean sphere of dimensionn − 1. An easy estimate shows that there exists r0 > 0 such that for anyradius r > r0

vol(Bnr ) ≥ Aneknr,

60

for some constants An, kn depending only on the dimension n. From this,we can argue as in Proposition 5.1.1 and obtain that

Proposition 5.4.1. For any compact hyperbolic manifold M , there exist con-stants C1, C2 depending only on the dimension of M such that

sys1(M) ≤ C1 log(vol(M)) + C2.

When the hyperbolic manifold M is non-compact, the argument of theinjectivity radius does not apply, and it is necesary to be careful with thecusps of the manifold. Recent work by M. Gendulphe [17] implies that forany non-compact hyperbolic manifold with finite volume we have

sys1(M) ≤ 2 log(vol(M)) + d1,

with d1 an explicit constant depending on the dimension of M . It is worthto note that in dimension n = 3, if the volume of M is large enough G. S.Lakeland and C. J. Leininger proved that

sys1(M) ≤ 4

3log(vol(M)) + d2,

for some constant d2 [25]. The constant 43

is particularly interesting becauseit appears in the lower bound in dimension 2.

As it was mentioned in the introduction of this thesis, the existence of com-pact hyperbolic 3-manifolds with arbitrary short closed geodesic followsfrom Thurston’s hyperbolic Dehn surgery theorem [39, Thm. 5.8.2]. In2006, I. Agol constructed compact hyperbolic 4-manifolds with arbitraryshort systole [1], and M. Belolipetsky and S. Thomson in 2011 adaptedAgol’s method to any dimension to solve the problem in the general case[4].

On the other side, it is natural to expect that congruence coverings of arith-metic hyperbolic n-manifolds attain the logarithmic bound for their sys-tole. In fact, comparing the geometry of the fundamental group of a com-pact hyperbolic manifold with the geometry of Hn it is possible to show

61

that there are constants C3, C4 depending only on n such that

sys1(MI) ≥ C3 log(vol(MI))− C4.

The proof can be found in [19, Prop. 10] (cf. [18, 3.C.6]). Although itshows that the systole of congruence coverings is bounded by a logarith-mic function of the volume, the proof uses gross comparisons between thegeometry of Γ(I) and Hn, and it does not give a precise value for the con-stants C3 and C4.

In dimension 3, M. Katz, M. Schaps and U. Vishne also found a preciselower bound for the systole [23, Thm. 1.8].

Theorem 5.4.2. Let Γ be an arithmetic co-compact torsion free subgroup of Isom+(H3).Then, the principal congruence subgroups of Γ satisfy

sys1(MI) ≥2

3log(vol(MI))− c,

where MI = Γ(I)\H3 and c is a constant that does not depend on MI .

62

CHAPTER 6

SYSTOLE OF CONGRUENCE COVERINGS OFARITHMETIC HYPERBOLIC MANIFOLDS

In this chapter we will bound by below the systole of congruence cov-erings of arithmetic hyperbolic manifolds of the first type associated toprime ideals. If Γ ⊂ Isom+(Hn) is such an arithmetic subgroup constructedfrom SO(1, n), we will prove that the congruence subgroups Γ(I) satisfy

sys1(MI) ≥4

n(n+ 1)log(vol(MI))− d1

for some constant d1 independent of I , whereMI = Γ(I)\Hn (Theorem A’).However, if Γ is constructed from Spin(1, n) we obtain a stronger lowerbound

sys1(MI) ≥8

n(n+ 1)log(vol(MI))− d2,

where d2 is a constant independent on I (Theorem A). We also prove thatthe constant 8

n(n+1)is sharp (Theorem B) and we give some applications.

In order to bound sys1(MI), we will follow the same path as in the Buser-Sarnak construction:

Step 1. We find an invariant by conjugation which measures the displace-ment of elements in Γ acting on Hn.

63

Step 2. When C ∈ Γ(I), we bound by below this invariant in terms of thenorm of the ideal I .

Step 3. We relate vol(MI) with the norm of the ideal I .

When we consider Γ as a subgroup of SO(1, n)◦, the invariant consideredis the trace. When Γ is a subgroup of Spin(1, n) the invariant consideredis the real part of the elements in Spin(1, n). Steps 2 and 3 are similar inboth cases: In Step 2 we use some ideas from the work by Katz, Schapsand Vishne [23]. In Step 3 we use the well-known formulas for the orderof orthogonal groups over finite fields.

6.1 Length-trace inequality for SO(1, n)◦

Among different conjugation invariants in a linear group the trace is one ofthe easiest to compute. We will see in this section that forC ∈ SO(1, n)◦ thetrace tr(C) contains information about the translation of points in Hn byC.

Proposition 6.1.1. For any C ∈ SO(1, n)◦ and any x ∈ Hn we have

|tr(C)| ≤ 2 cosh ρ+ n− 1 ≤ eρ + n (6.1)

with ρ = dHn(x,Cx).

Proof. We first consider the particular case when x = e1 andCx = cosh(ρ)e1 + sinh(ρ)e2 for some ρ ≥ 0. Consider the matrix Aρ ∈SO(1, n)◦ given by

Aρ =

cosh(ρ) sinh(ρ)

sinh(ρ) cosh(ρ)0

0 In−1×n−1

.

Since the curve α : R → Hn defined by α(t) = cosh(t)e1 + sinh(t)e2 is ageodesic in Hn and C(e1) = Aρ(e1) = α(ρ) we have that d(e1, Aρ(e1)) =

64

d(α(0), α(ρ)) = ρ.

Now, A−1ρ C fixes the point e1 and so there exists an orthogonal matrix

B = (bij) of size n× n such that

C =

cosh(ρ) sinh(ρ)

sinh(ρ) cosh(ρ)0

0 In−1×n−1

( 1 01×n

0n×1 B

).

A direct computation shows that

tr(C) = cosh ρ+ b1,1 cosh ρ+ b2,2 + · · ·+ bn,n.

Since the matrix B = (bij) is orthogonal, the entries bij have norm less orequal to 1 for any i, j = 1, . . . , n. Therefore

|tr(C)| ≤ 2 cosh ρ+ n− 1.

Suppose now that x is any point in Hn and C ∈ SO(1, n)◦. Since Hn is abihomogeneous space, if we take ρ = d(x,Cx) there exists an element T ∈SO(1, n)◦ such that T (x) = e1 and TC(x) = α(ρ). Then TCT−1 ∈ SO(1, n)◦

and TCT−1(e1) = α(ρ). The result now follows from the previous caseand the invariance of the trace by conjugation. The right hand side ofInequality (6.1) follows from the definition of cosh(ρ).

6.2 A trace estimate for congruence subgroups

In this section we will study the principal congruence subgroups of Γ =

SOf (Ok), where k is a totally real number field with d = [k : Q], and f isan admissible quadratic form defined over k such that in a suitable basisf has the form

f = a1x21 − a2x

22 − · · · − an+1x

2n+1

with ai ∈ Ok. It means that ai > 0 for i = 1 . . . , n+1 and for any non-trivialplace σ : k → R we have σ(a1) > 0 and σ(ai) < 0 for i = 2, . . . , n+ 1.

65

We would like to find a lower bound for the trace of the elements of Γ(I) interms of the norm N(I) of the ideal I , which is by definition the cardinalityof the quotient ring Ok/I . To do so we will fix the representations for thesubgroups Γ(I) and we use some ideas from Katz, Schaps and Vishne in[23].

We can represent the orthogonal group SOf (R) as the subgroup of SLn+1(R)

given by

SOf (R) = {C ∈ SLn+1(R) | CtJC = J}

with

J =

a1 0 · · · 0

0 −a2 · · · 0...

... . . . ...0 0 · · · −an+1

.

With this representation SOf (Ok) = SOf (R)◦ ∩ SLn+1(Ok). In other words

Γ = {(cij) ∈ SOf (R)◦ | cij ∈ Ok}.

For any ideal I ⊂ Ok the principal congruence subgroup Γ(I) correspondsto the kernel of the projection map Γ

πI−→ SLn+1(Ok/I), which is equal tothe group

Γ(I) = {(cij) ∈ Γ | cii − 1 ∈ I, cij ∈ I for i 6= j}.

By a direct computation, the equation CtJC = J applied to the canonicalbasis of Rn+1 implies that

a1c21,1 − · · · − anc2

n,1 − an+1c2n+1,1 = a1, (6.2)

a1c21,j − a2c

22,j · · · − ajc2

j,j − · · · − anc2n,j − an+1c

2n+1,j = −aj, for j = 2, . . . , n.

(6.3)As we remarked in Section 4.7, Γ embeds into SO(1, n)◦ via an isomor-phism SOf (R)◦ ' SO(1, n)◦ which is a trace-preserving map, therefore wecan bound the traces of elements in Γ(I) directly in SOf (R)◦. To do so weuse Equations (6.2) and (6.3).

66

Lemma 6.2.1. Let C = (cij) be an element in SOf (R)◦ and write ci,i = 1 + yi.

1. For every place σ 6= id and i = 1, . . . , n + 1, we have |σ(ci,i)| ≤ 1, andhence |σ(yi)| ≤ 2.

2. If C ∈ Γ(I) we have

2an+1∑i=1

yi ∈ I2, (6.4)

where a =∏n+1

i=1 ai. In particular, |N(∑n+1

i=1 yi)| ≥ N(I)2

2d N(a)2.

Proof. By hypothesis we have that σ(a1) > 0 and σ(aj) < 0 for j = 2, . . . , n+

1. Applying σ to Equation (6.2) we obtain

σ(a1)σ(c1,1)2 ≤ σ(a1)σ(c1,1)2 − · · · − σ(aj)σ(cj,1)2 − · · · − σ(an)σ(cn,1)2

− σ(an+1)σ(cn+1,1)2

= σ(a1)

Hence |σ(c1,1)| ≤ 1. By the same argument, if we apply σ in Equation (6.3)we obtain the corresponding fact for i = 2, . . . , n + 1 and we have provedthe first part.

To prove Part 2 we replace c1,1 = 1 + y1 in Equation (6.2) and we obtain

2a1y1 + y21 − a2c

22,1 − · · · − anc2

n,1 − an+1c2n+1,1 = 0. (6.5)

If C ∈ Γ(I) then y1 ∈ I and ci,1 ∈ I for i = 2, . . . , n+ 1. From Equation (6.5)it follows that 2a1y1 ∈ I2. By the same argument replacing ci,i = 1 + yi inEquation (6.2) we obtain that 2aiyi ∈ I2 for i = 2, . . . , n + 1. In particular,2ayi ∈ I2 for any i = 1 . . . , n+ 1 and hence their sum lies in I2.

This lemma allows us to obtain a lower bound for the trace of elements inΓ(I) in terms of the norm of the ideal I :

Proposition 6.2.2. If C ∈ Γ(I) is not the identity we have

|tr(C)| ≥ N (I)2

22d−1(n+ 1)d−1∏n+1

i=1 N(ai)− n− 1.

67

Proof. By the definition of the norm of elements ofO and Part 1 of Lemma6.2.1 we have∣∣∣∣∣N

(n+1∑i=1

yi

)∣∣∣∣∣ =

∣∣∣∣∣n+1∑i=1

yi

∣∣∣∣∣∣∣∣∣∣ ∏σ 6=id

σ

(n+1∑i=1

yi

)∣∣∣∣∣ ≤ 2d−1 (n+ 1)d−1

∣∣∣∣∣n+1∑i=1

yi

∣∣∣∣∣.

This fact together with Part 2 of Lemma 6.2.1 implies∣∣∣∣∣n+1∑i=1

yi

∣∣∣∣∣ ≥ N (I)2

22d−1(n+ 1)d−1∏n+1

i=1 N(ai).

Now, since tr(C) = n + 1 +∑n+1

i=1 yi we have the following lower boundfor the trace of the elements in Γ(I)

|tr(C)| ≥

∣∣∣∣∣n+1∑i=1

yi

∣∣∣∣∣− n− 1 (6.6)

≥ N (I)2

22d−1(n+ 1)d−1∏n+1

i=1 N(ai)− n− 1. (6.7)

6.3 A first lower bound for sys1(MI)

Now we are able to give examples with an explicit value of the constant Cin Inequality (1.4) in dimensions n ≥ 4.

Theorem A’. Let Γ be an arithmetic subgroup of SO(1, n)◦ of the first type de-fined over a totally real number field k. Then any sequence of congruence sub-groups Γ(I) with I being a prime ideal and N(I) going to infinity eventuallysatisfy

sys1(MI) ≥4

n(n+ 1)log(vol(MI))− d1,

where MI = Γ(I)\Hn and d1 is a constant independent of I .

68

Proof. We can assume that Γ = SOf (Ok) ∩ SOf (R)◦ where f has the diago-nal form f = a1x

21−a2x

22−· · ·−an+1x

2n+1 with ai ∈ Ok such that for any non

trivial place σ : k → R we have σ(a1) > 0 and σ(ai) < 0 for i = 2, . . . , n+ 1.These assumptions may require passing to a finite sheet covering, whichis compatible with the statement of the theorem.

Let γ be a non-homotopically trivial closed geodesic on MI = Γ(I)\Hn.We recall that there exist x ∈ Hn and C ∈ Γ(I) such that the only geodesicjoining x and Cx is a lifting of γ and `(γ) = d(x,Cx). Since γ is non-homotopically trivial, Cx 6= x; in particular C is not the identity elementin Γ(I). For N(I) large enough Proposition 6.2.2 shows that |tr(C)| −n > 0

and then we can apply Proposition 6.1.1 to obtain the bound

`(γ) ≥ log(|tr(C)| − n).

Applying the lower bound for the trace of elements in Γ(I) given by Propo-sition 6.2.2 we conclude that

sys1(MI) ≥ log

(N (I)2

22d−1(n+ 1)d−1∏n+1

i=1 N(ai)− 1− 2n

)(6.8)

≥ log(N(I)2

)− log

(22d(n+ 1)d−1

n+1∏i=1

N(ai)

)(6.9)

if N(I)2 ≥ (1 + 2n)22d(n+ 1)d−1∏n+1

i=1 N(ai).

Now, for N(I) large enough the elements ai, i = 1, . . . , n do not belong toI and so the quadratic form f = a1x

21 − a2x

22 − · · · − an+1x

2n+1 defined over

the finite field FN(I) = Ok/I is non-degenerated.

The formula for the order of the orthogonal group over finite fields givesthe bound | SOf (FN(I))| ≤ N(I)

n(n+1)2 (see e.g. [41, Sec. 3.7.2]). Since [Γ :

Γ(I)] ≤ | SOf (FN(I))| Equation (6.9) implies that

sys1(MI) ≥4

n(n+ 1)log ([Γ : Γ(I)])− log

(22d(n+ 1)d−1

n+1∏i=1

N(ai)

).

69

The result now follows from the fact that vol(MI) = ν[Γ : Γ(I)], whereν = vol(Γ\Hn).

6.4 The displacement of elements in Spin(1, n)

acting on Hn

In this section we explore the action of Spin(1, n) on Hn and we prove thatthe real part (see the definition below) of elements in Spin(1, n) containsinformation about the displacement of elements in Spin(1, n) acting on Hn.

Since we will focus our attention on the group Spin(1, n), we can fix thereal vector space E = Rn+1, the quadratic form q = x2

1−x22−· · ·−x2

n+1 andthe canonical basis {e1, e2, . . . , en+1} of Rn+1. The Clifford algebra C (q) canbe then described as the R-algebra generated by e1, e2, . . . , en+1 with therelations e2

1 = 1, e2j = −1 for j = 2, . . . , n+ 1 and eiej = −ejei for i 6= j. The

following lemma will often allow us to simplify the situation by conjugat-ing in the group Spin(1, n).

Definition 6.4.1. For s = ΣM∈PnsMeM in C (q) we define the real part of sas sR := s∅.

Lemma 6.4.2. The real part of the elements of Spin(1, n) is conjugation invari-ant.

Proof. Consider the faithful representation L : Spin(1, n)→ GL2n(R) givenby the action of Spin(1, n) on C (q) by the left multiplication Ls(x) = sx.For the basis element eM , the coefficient in eM of Ls(eM) is equal to sR,hence the associated matrix of Ls in this basis has all its entries in the maindiagonal equal to sR. This implies that the trace of Ls is equal to 2n+1sR,and since the trace of matrices is invariant by conjugation, it concludes theproof.

70

Now we can relate the displacement and the real part of elements in Spin(1, n).The main step is the following lemma.

Lemma 6.4.3. Let s ∈ Spin(1, n) and ϕs its image under ϕ in SO(1, n)◦. IfA = (ai,j)i,j=1,...,n+1 represents ϕs in the basis {e1, . . . , en+1}, we have

cosh(d(e1, ϕs(e1)) = a1,1.

In particular, if s =∑M

sMeM , then cosh(d(e1, ϕs(e1)) =∑M

s2M .

Proof. We know that the stabilizer of the point e1 = (1, 0, . . . , 0) ∈ Hn forthe action of SO(1, n)◦ on Hn is given by the subgroup of matrices of theform

B =

1 0 0 · · · 0

0 b1,1 b1,2 · · · b1,n

......

... . . . ...0 bn,1 bn,2 · · · bn,n

, (6.10)

where B = (bi,j) is an n× n orthogonal matrix.

Now, ϕs(e1) = A(e1) = (a1,1, . . . , an+1,1) and we can find an orthogonalmatrix B such that

B

a2,1

a3,1

...an+1,1

=

0

0...c

, where c2 = a22,1 + · · ·+ a2

n+1,1.

Then, choosing B as in (6.10) we have

cosh(d(e1, ϕs(e1)) = cosh(d(Be1, Bϕs(e1))

= cosh(d(e1, a1,1e1 + cen+1)).

71

It is well known that the curve α(t) = cosh(t)e1 + sinh(t)en+1 is a mini-mizing geodesic in Hn. Since a2

1,1 − c2 = 1 there exists t ∈ R such thatcosh(t) = a1,1 and sinh(t) = c, therefore

cosh(d(e1, a1,1e1 + cen+1)) = cosh(d(α(0), α(t)))

= cosh(t)

= a1,1.

Now, if s =∑M

sMeM then se1s∗ =

∑M,N

sMsNeNe1e∗M . Note that if N 6= M

the product eNe1e∗M is a basis element different from e1; if N = M and eM

contains e1 in its product then eNe1e∗M = −e1eMe

∗M = e1; and ifN = M and

eM does not contain e1 in its product then eNe1e∗M = e1eNe

∗N = e1. With all

this we deduce that the coordinate e1 of se1s∗, which is equal to a1,1 agrees

with∑M

s2M .

We conclude this section by relating the displacement of an element s ∈Spin(1, n) with its real part. This relation is a generalization of the well-known fact that in dimensions 2 or 3 the displacement of an isometry is re-lated to the trace of the matrix in SL(2,R) or SL(2,C), see e.g. [2, Chap. 7].

Proposition 6.4.4. For any s ∈ Spin(1, n) with |sR| ≥ 1 we have

d(x, ϕs(x)) ≥ 2 log(|sR|).

Proof. Since the real part is conjugation invariant, by conjugating by anelement in Spin(1, n) we can suppose that x = e1. In this case, if s =∑M

sMeM by Lemma 6.4.3 we have

cosh(d(e1, ϕs(e1)) =∑M

s2M ≥ s2

R.

Hence d(e1, ϕs(e1)) ≥ arccosh(s2R) ≥ 2 log(|sR|).

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6.5 Lower bound for the displacement ofcongruence subgroups

In this section we will study the displacement of the principal congruencesubgroups Γ(I) of Γ = Spinf (Ok) in terms of the ideal I . As in Section 6.2the main ideas are inspired by [23] where the authors study the systole ofcompact hyperbolic surfaces and 3-manifolds.

We will start by giving a representation of the principal congruence sub-groups of Γ with f an admissible quadratic form. We will use the mul-tiplicative structure of the algebra C (f,R) to describe an embedding ofSpin(f,R) into GLm for somem. Choose an orthogonal basisB = {e1, . . . , en+1}with respect to f . The Clifford algebra C (f,R) is a real vector space of di-mension 2n+1 with basis {eM}M∈Pn+1 and the group Spin(f,R) acts on itby left multiplication. For any s ∈ Spin(f,R), the linear map Ls(x) = sx,x ∈ C (f,R), belongs to GL(C (f,R)) ' GL2n+1(R) and so we have a lin-ear representation L : Spin(f,R) → GL2n+1(R). If s =

∑|M | even

sMeM with

sM ∈ R, then Ls ∈ GL2n+1(Ok) if and only if all sM ∈ Ok. We then obtainthat

Γ =

{s =

∑|M | even

sMeM | sM ∈ Ok, sEs∗ = E and ss∗ = 1

}.

Note that for any s ∈ Γ the coordinates of seis∗ lie in Ok for any i =

1, . . . , n+1, therefore ϕ(Γ) ⊂ SOf (Ok)∩SOf (R)◦. Since SOf (Ok)∩SOf (R)◦

andϕ(Γ) are lattices in SOf (R)◦ the imageϕ(Γ) has finite index in SOf (Ok)∩SOf (R)◦ and these subgroups are commensurable.

Now, for an ideal I ⊂ Ok the principal congruence subgroup Γ(I) cor-responds to the kernel of the projection map Γ

πI−→ GL2n+1(Ok/I), whichgives

73

Γ(I) =

{s =

∑|M | even

sMeM ∈ Γ | sM ∈ I for M 6= ∅ and sR − 1 ∈ I

}.

By Proposition 6.4.4 one can find a lower bound for the displacement ofelements in Γ(I) from a lower bound for the real part of its elements. Thecondition ss∗ = 1 implies that

∑|M | even

s2Mf(eM) = 1, (6.11)

where forM = {iv1 , . . . , ivk}we denote by f(eM) the product f(eiv1 ) · · · f(eivk ),which lies in Ok.

If s ∈ Γ(I), we obtain that s2R − 1 ∈ I2. By writing sR = y0 + 1 with y0 ∈ I ,

we get 2y0 ∈ I2. From this we can obtain a lower bound for |sR| in termsof the norm of the ideal I .

Lemma 6.5.1. For any non-trivial s ∈ Γ(I), we have |sR| ≥ N(I)2

22d−1 − 1.

Proof. By applying any non-trivial embedding σ to Equation (6.11), sincefσ is positive-definite, we obtain

σ(sR)2 ≤∑

|M | evenσ(sM)2fσ(eM) = 1 (6.12)

By replacing σ(sR) = σ(y0)+1 in Equation (6.12), we obtain that |σ(y0)| ≤ 2.Observe that σ(y0) 6= 0, since otherwise σ(sR) = 1 and Equation (6.12)would imply σ(sM) = 0 for any M 6= ∅, so by injectivity of σ we wouldthen have s = 1.

Since 2y0 ∈ I2, we have N(y0) ≥ N(I)2

2d. By definition, N(y0) is equal to the

product |y0|∏σ 6=1

|σ(y0)| and so |y0| ≥ N(I)2

22d−1 . This implies that

|sR| ≥ |y0| − 1 ≥ N(I)2

22d−1− 1.

74

This, together with the results obtained in the previous section, allows usto relate the displacement of elements of Γ(I) with the norm N(I).

Proposition 6.5.2. If I ⊂ Ok is an ideal with norm N(I) ≥ 2d, then for anynon-trivial s ∈ Γ(I) and x ∈ Hn we have

d(x, ϕs(x)) ≥ 4 log(N(I))− 4d log(2).

Proof. The condition N(I) ≥ 2d implies that N(I)2

22d−1 − 1 ≥ N(I)2

22dand so |sR| ≥

N(I)2

22d≥ 1. The result follows from Proposition 6.4.4 and Lemma 6.5.1.

This proposition shows, in particular, that for ideals with norm large enoughthe congruence subgroup Γ(I) acts without fixed points and so the quo-tient MI = Γ(I)\Hn is a hyperbolic manifold.

6.6 Proof of the main theorem

In the proof of Theorem A we will use some properties of the spin groupSpinf , with f being a quadratic form defined over a finite field. We startwith a lemma [31, Lem. 3.7].

Lemma 6.6.1. Let F be a finite field, n ≥ 2 and f = b1x21 + · · · + b2

n+1x2n+1 be a

quadratic form with bi ∈ F×, then f is non-degenerated and isotropic.

Proof. The fact that f is non degenerated follows from the diagonal formof f . For the second part it is enough to prove that the quadratic formg = b1x

21 + b2x

22 + b3x

23 is universal. For c ∈ F the sets A = {b1 + b2y

2 | y ∈ F}and B = {c − b3z

2 | z ∈ F} have the same cardinality, equal to |F|+12

,therefore A ∩B 6= ∅ and there exist y2 and y3 in F such that

b1y21 + b2y

22 + b3y

23 = c

with y1 = 1.

We can now put all the pieces together and prove the main result.

75

Theorem A. Let Γ be an arithmetic subgroup of Isom+(Hn) of the first typedefined over a totally real number field k. Then for any sequence of prime idealsI ⊂ Ok the principal congruence subgroups Γ(I) satisfy

sys1(MI) ≥8

n(n+ 1)log(vol(MI))− d2,

where MI = Γ(I)\Hn and d2 is a constant independent of I .

Proof. Without loss of generality we can assume that there exists an ad-missible quadratic form f of the form f = a1x

21 − a2x

22 − · · · − an+1x

2n+1

with ai ∈ Ok and Γ = Spinf (Ok). These assumptions may require passingto a finite sheet covering, which is compatible with the statement of thetheorem.

If I ⊂ Ok is a prime ideal we denote by FN(I) the fieldOk/I and f = a1x21−

a2x22 − · · · − an+1x

2n+1 the projection of f to a form over FN(I). If the norm

of the ideal I is big enough then 2 and the ai with i = 1, . . . , n + 1 do notbelong to I and so char(FN(I))6= 2, f is non-degenerate and it is an isotropicform by the previous lemma. Since char(FN(I))6= 2, we saw in Section 4.2that |SOf (FN(I))| = |Spinf (FN(I))| and we know that the cardinality of thespecial orthogonal group SOf (FN(I)) is bounded by above by N(I)

n(n+1)2

(see e.g. [41, Sec. 3.7.2]). Therefore

[Γ : Γ(I)] ≤ |Spinf(FN(I)

)| = |SOf

(FN(I)

)| ≤ N(I)

n(n+1)2 .

Now, if α is a non-contractible closed geodesic on MI = Γ(I)\Hn, there isan element s ∈ Γ(I) and a point x ∈ Hn such that d(x, ϕs(x)) = `(α). ByProposition 6.5.2 we conclude that

sys1(MI) ≥8

n(n+ 1)log([Γ : Γ(I)])− 4d log(2).

Since vol(MI) = [Γ : Γ(I)]vol(M), where M = Γ\Hn, it follows that ifthe norm of the ideal I is large enough, we obtain the desired inequalitywith d2 = 4d log(2) + 8

n(n+1)log(vol(M)). The systole for the other I can be

compensated by enlarging the constant.

76

6.7 Sharpness of the lower bound

In this section we would like to prove that the constant 8n(n+1)

in the sys-tole growth in Theorem A is the best possible. The next theorem has beendeveloped in collaboration with Cayo Dória. The proof uses the result ofS. Makisumi in dimension 2 (Theorem 5.3.3).

Theorem B. Let Γ be an arithmetic subgroup of the first type in Isom+(Hn)

defined over a totally real number field k. Then, up to passing to a commensu-rable group, for any sequence of prime ideals I ⊂ Ok the principal congruencesubgroups Γ(I) satisfy

sys1(MI) ≤8

n(n+ 1)log(vol(MI)) + d3

where MI = Γ(I)\Hn and d3 is a constant independent of I .

Proof. Passing to a commensurable group, we assume that Γ = Spinf (Ok),with f being an admissible quadratic form such that in the basis {e1, . . . , en+1}has the diagonal form a1x

21 − a2x

22 − · · · − an+1x

2n+1 with ai ∈ Ok. This im-

plies that ai > 0 and for any non-trivial embedding σ : k → R, σ(a1) > 0

and σ(ai) < 0 for i = 2, . . . , n+ 1.

LetE ′ ⊂ E be the 3-dimensional subspace generated by {e1, e2, e3} and notethat the restriction f ′ : E ′ → k of f to E ′ has signature (1, 2). In this casethe group Γ′ = Spinf ′(Ok) is an arithmetic lattice of Spinf ′(R) and we havea natural inclusion Spinf ′(Ok) ↪→ Spinf (Ok). Consider now an isometricembedding of H2 to Hn equivariant by the respective actions of Γ′ and Γ.For any ideal I ⊂ Ok we obtain a totally geodesic embedding

SI ↪→MI

where SI = Γ′(I)\H2. In particular, for any I

sys1(MI) ≤ sys1(SI).

77

On the other hand, the even Clifford algebra C +(f ′) is the quaternion al-gebra

A = k[i, j|i2 = a, j2 = b, ij = −ji],

with a = a1a2 and b = a1a3 ∈ Ok. Moreover, the group Γ′ coincide with thegroup of units of the order A(Ok) and Γ′(I) is the kernel of the projectionmap A(Ok) → A(Ok/I) for any ideal I . With this identification Γ′ actson both of the models of the hyperbolic plane: on the upper half spacemodel Γ′ acts via the quaternion algebra A and on the hyperboloid modelit acts via Spinf ′(R). It is known that there exists an isometry betweenthis two models which is equivariant with respect to the actions of Γ′ [21,Proposition 5.3], therefore applying Theorem 5.3.3 to the sequence SI forsome ε > 0 we get

sys1(SI) ≤4

3log(area(SI)) + ε

when N(I) → ∞. To finish the proof, we relate area(SI) with N(I) follow-ing the same arguments as in [3, Proposition 3.2]. For N(I) large enoughvol(MI) = ν| Spinf (Ok/I)|, where ν = vol(Γ\Hn). As we saw in the proofof Theorem A for prime ideals I with large norm the projection of thequadratic form f to the finite field Ok/I is non-degenerate and the order| Spinf (Ok/I)| grows as N(I)

n(n+1)2 . This implies that the quotients

vol(MI)

ν N(I)n(n+1)

2

andarea(SI)

µN(I)3

tend to constants when N(I)→∞, where µ = area(Γ′\H2). It follows that

log(area(SI)) ≤6

n(n+ 1)log(vol(MI)) + d′3

for some constant d′3. Therefore

sys1(MI) ≤8

n(n+ 1)log(vol(MI)) +

(4d′33

+ ε

)when N(I)→∞.

78

6.8 Applications

Historically, sequences of congruence coverings have been used in manydifferent contexts. We would like to apply Theorem A to improve someresults obtained in recent years.

6.8.1 Systolic genus of hyperbolic manifolds

The first application concerns to the systolic genus of arithmetic hyper-bolic manifolds studied by M. Belolipetsky in [3]. If we denote by Sg aRiemann surface of genus g ≥ 1, then the systolic genus of a Riemannianmanifold M is defined by

sysg(M) = min{g | π1(M) contains π1(Sg)}.

The main result in [3] relates the systolic genus sysg(M) of a hyperbolicmanifold M with the systole sys1(M).

Theorem 6.8.1. [3, Theorem 5.1] Let M be a closed n-dimensional hyperbolicmanifold. For any ε > 0, assuming that sys1(M) is sufficiently large, we have

sysg(M) ≥ e( 12−ε)sys1(M).

Concerning the congruence coverings in dimension n ≥ 3, the followingresult is proved in [3]:

Proposition 6.8.2. [3, Proposition 5.3]. Let Γ be a fundamental group of a closedarithmetic hyperbolic manifold of dimension n ≥ 3.

(A) There exists constant D > 0 such that for a decreasing sequence Γi < Γ

of congruence subgroups of Γ, the corresponding quotient manifolds Mi =

Γi\Hn satisfy

log sysg(Mi) & D log(vol(Mi)), as i→∞.

(B) If Γ is of the first type, the sequence of principal congruence subgroups Γ(I)

satisfysysg(MI) . vol(MI)

6n(n+1)

., as N(I)→∞,

79

where MI = Γ(I)\Hn.

We recall that for two positive functions f(x) and g(x), the relation f(x) &

g(x) means that for any ε > 0 there exists x0 depending on ε such thatf(x) ≥ (1− ε)g(x) for all x ≥ x0.

The explicit constant D = 13

is known for dimension n = 3 [3, Proposition3.1], but in higher dimensions no explicit value of this constant was knownso far. In this respect, we can apply Theorem A to give a quantitativeversion of this result.

Proposition 6.8.3. Let Γ be the fundamental group of a closed arithmetic hyper-bolic manifold of the first type of dimension n ≥ 3. Then Γ has a sequence ofcongruence subgroups Γi such that the quotient manifolds Mi = Γi\Hn satisfy

log sysg(Mi) &4

n(n+ 1)log(vol(Mi)), as i→∞.

Proof. Consider the principal congruence subgroups ΓI < Γ with I rang-ing over prime ideals. By Theorem A,

sys1(MI) &8

n(n+ 1)log(vol(MI)), as N(I)→∞

The result now follows from Theorem 6.8.1.

Joining together Theorem 6.8.1 with Part (B) of Proposition 6.8.2 we obtainan upper bound for the systole of the manifolds considered in Theorem A.

Corollary 6.8.4. Let n ≥ 3 and let Γ be an arithmetic subgroup of Spin(1, n) ofthe first type defined over a totally real number field k. Then for any sequence ofprime ideals I ⊂ Ok the principal congruence subgroups Γ(I) satisfy

8

n(n+ 1)log (vol(MI)) . sys1 (MI) .

12

n(n+ 1)log (vol(MI)) ,

as N(I)→∞, where MI = Γ(I)\Hn.

Note that the upper bound in the corollary is not as sharp as Theorem Bbut is obtained by different methods.

80

6.8.2 Homological Codes

Motivated by a question of Zémor [42], L. Guth and A. Lubotzky con-structed in 2013 a certain class of homological codes using congruencecoverings of arithmetic hyperbolic 4-dimensional manifolds [19]. For thedefinition of homological codes and the details of the construction usinghyperbolic manifolds we refer the reader to [19] and the references therein.

According to the known examples, Zémor [42] asked if it is true that every[[n, k, d]] homological quantum code satisfies the inequality kd2 ≤ n1+o(1).E. Fetaya [15] proved that it holds for surfaces but L. Guth and A. Lubotzky[19] gave counterexamples in dimension 4. The construction comes fromcongruence coverings of a compact 4-dimensional arithmetic hyperbolicmanifold. The main result in [19] can be stated in the following way.

Theorem 6.8.5. [19, Theorem 1]. Let M be a compact arithmetic hyperbolic4-manifold. There exist constants ε, ε1, ε2 > 0 such that for a sequence of congru-ence coverings MI with triangulations X the associated homological quantumcodes constructed in C2 (X,Z2) = C2 (X,Z2) are [[n, ε1n, n

ε2 ]] codes and satisfy

kd2 ≥ n1+ε.

With respect to Zémor’s question, it could be interesting to obtain an ex-plicit value of the constant ε > 0 in the previous result. Since the construc-tion in [19] makes use of congruence coverings of arithmetic hyperbolic4-manifolds, we can use Theorem A to give an explicit value of this con-stant.

LetM be a compact arithmetic hyperbolic 4-manifold of the first type withΓ = π1(M) as in Theorem A. The compactness of M implies that Γ is de-fined over a totally real number field k 6= Q and that MI is compact forany ideal I ⊂ Ok. Therefore, the injectivity radius of MI is equal to sys1(MI)

2

and Theorem A implies that for prime ideals I ⊂ Ok with sufficiently largenorm, MI satisfy

inj(MI) ≥1

5log(vol(MI))− c,

81

for some constant c independent ofMI . By using this bound in the proof ofTheorem 6.8.5, we obtain that the distance of the codes in this constructionsatisfies

d ≥ c1n0.2,

for some positive constant c1. It is known that d = O(n0.3) [19, Remark 20],hence this bound is quite close to the optimal one. To conclude with theestimate in Theorem 6.8.5 for ε > 0, we recall that the dimension k of thecode satisfies

k ≥ c2n,

for some positive constant c2 if the norm of the ideal I is sufficiently large[19, Theorem 6]. Therefore, we obtain that for prime ideals I with suffi-ciently large norm, the homological codes constructed from MI satisfy

kd2 ≥ c3n1+0.4,

for some positive constant c3.

82

CHAPTER 7

SYSTOLE OF CONGRUENCE COVERINGS OFHILBERT MODULAR VARIETIES

In this chapter we will prove that any sequence of congruence coveringsMI of a Hilbert modular variety satisfies

sys1(MI) ≥4

3√n

log(vol(MI))− d2, (7.1)

for some constant d2 independent of I (Theorem C). Furthermore, we willprove that sys1(MI) grows at most logarithmically with the volume of MI ,and that the constant 4

3√n

in Inequality (7.1) cannot be improved in general(Theorem D).

In this case we will follow the path described in the introduction of Chap-ter 6 and only steps 2 and 3 will needed. In Step 2 we use similar ideasas in the hyperbolic case adapted to the context. In Step 3 we give explicitbounds relating vol(MI) and the norm of the ideal I .

7.1 Hilbert modular varietes

We recall the definition of a Hilbert modular variety, which generalize theHilbert modular surfaces presented in Example 3.6.2 and Example 4.6.4.

83

We refer the reader to [16] for more details about this topic.

Let k be a totally real number field of degree n, and let σ1, . . . , σn denotethe n embeddings of k into the real numbers R. By Proposition 4.5.4 thegroup PSL2(Ok) becomes an arithmetic lattice of the semisimple Lie group(PSL2(R))n via the map ∆(B) = (σ1(B), . . . , σn(B)), where σi(B) denotesthe matrix obtained by applying σi to the entries ofB. Via this embedding,PSL2(Ok) acts on the n-fold product of hyperbolic planes (H2)n with finitecovolume (see [32, Proposition 5.5.8]). The quotientMk = PSL2(Ok)\(H2)n

is called a Hilbert modular variety and the group Γ = PSL2(Ok) is called aHilbert modular group.

The principal congruence subgroups of Γ are given by

Γ(I) =

{(a b

c d

)∈ SL2(Ok) | a− 1, d− 1, b, and c lie in I

}/{1,−1}.

for any ideal I ⊂ Ok, and we will denote by MI the corresponding con-gruence coverings Γ(I)\(H2)n of Mk.

7.2 Upper bound for the systole growth of MI

As was explained in Section 5.2, if Λ is any discrete group of isometries ofH2 acting freely on H2, every hyperbolic element γ ∈ Λ produces a non-contractible closed geodesic on H2/Λ. We can use this idea to see that thequotients MI which we are interested in have closed geodesics, and sub-sequently we find a upper bound for sys1(MI).

Suppose I ⊂ Ok is an ideal with N(I) > 2 and such that MI is a Rieman-nian manifold (see Corollary 7.3.3). The norm N(I) is a rational integerwith N(I) ∈ I , so if we take the matrix

B =

(1− N(I)2 N(I)

−N(I) 1

),

84

then B ∈ Γ(I) and |tr(σi(B))| > 2 for any i = 1, . . . , n. This means thatthe matrices σ1(B) = σ2(B) = · · · = σn(B) are hyperbolic and if we takeα to be the only geodesic in H2 fixed by B, the curve β = α × · · · × α isa geodesic in (H2)n that is fixed by (σ1(B), . . . , σn(B)), and β projects to anon-contractible closed geodesic β in MI . Note that this geodesic mightnot be the shortest one, so sys1(MI) ≤ `(β) =

√n`B, where `B denotes the

translation length of B along α.

We know that 2 cosh( `B2

) = |tr(B)| = N(I)2 − 2 < N(I)2, and so

sys1(MI) ≤ 4√n log(N(I)).

Now, as we will see in Section 7.4 there exists a constant Ck independentof I such that [Γ : Γ(I)] ≥ Ck N(I)3 (Lemma 7.4.2), and hence

sys1(MI) ≤4√n

3log([Γ : Γ(I)])− 4

√n

3log(Ck). (7.2)

This shows that sys1(MI) grows at most logarithmically in vol(MI) whenthe norm N(I) goes to infinity since vol(MI) = [Γ : Γ(I)]vol(M). We willsee in Section 7.6 that for some congruence coverings, Inequality (7.2) canbe improved significantly.

7.3 Distance estimate for congruence subgroups

In this section we obtain a lower bound for the displacement of Γ(I) actingon (H2)n in terms of the norm of the ideal I . As a consequence we havethat the congruence subgroups Γ(I) act freely on (H2)n when the norm ofthe ideal I is big enough and we obtain a lower bound for the length ofclosed geodesics in MI . As in the previous chapter, some of the ideas inthis section are inspired by [23].

We would like to make some comments about the notation we will use inthis section. Sometimes we are going to use the notationA or (σ1(A), . . . , σn(A))

85

indifferently for the same element in Γ or its image in (PSL2(R))n via themap ∆ defined in Section 7.1.

For our purpose, it is convenient to express any element A =

(a b

c d

)of

Γ in the form

A =

(x0 + x1 x2 + x3

x2 − x3 x0 − x1

),

where x0 = a+d2, x1 = a−d

2, x2 = b+c

2and x3 = b−c

2are elements of the field

k. We have x20 − x2

1 − x22 + x2

3 = 1 and we write y0 = x0 − 1.

Considering these notations, if I ⊂ Ok is an ideal and A ∈ Γ(I) then 2x0 −2 ∈ I and 2xi ∈ I for i = 1, 2, 3. In terms of fractional ideals it means thaty0, x1, x2 and x3 lie in I

2.

Lemma 7.3.1. If A ∈ Γ(I), then y0 ∈ I2

8. In particular, if y0 6= 0 then |N(y0)| ≥

18n

N(I)2.

Proof. We know that A ∈ Γ(I) implies x0 − 1, x1, x2, x3 ∈ I2. Now, by

replacing x0 = 1 + y0 in the equation x20 − x2

1 − x22 + x2

3 = 1 we obtain

2y0 = −y20 + x2

1 + x22 − x2

3 ∈I2

4.

Hence y0 ∈ I2

8.

Lemma 7.3.2. If A ∈ Γ(I) with y0 6= 0 then |tr(σj(A))| ≥ (N(I))4

2n − 2 for some

j ∈ {1, . . . , n}.

Proof. By definition, N(y0) =∏n

j=1 σj(y0), so by Lemma 7.3.1, for some

j ∈ {1, . . . , n}, we have |σj(y0)| ≥ N(I)2n

8. Therefore

|tr(σj(A))| = |2σj(x0)| = |2σj(y0) + 2| ≥ N(I)2n

4− 2.

With this we can guarantee the Riemannian structure for MI :

86

Corollary 7.3.3. For any ideal I ⊂ Ok with N(I) ≥ 4n the subgroup Γ(I) actsfreely on (H2)n and so MI = Γ(I)\(H2)n admits a structure of a Riemmanianmanifold with non-positive sectional curvature.

Proof. The element A = (σ1(A), . . . , σn(A)) ∈ Γ(I) has a fixed point on(H2)n if and only if σi(A) has a fixed point in H2 for any i = 1, . . . .n, but thishappens if and only if |tr(σi(A))| < 2 which, by Lemma 7.3.2, is impossibleif N(I) ≥ 4n.

Now observe that for i = 1, . . . , n and A ∈ Γ,

2|σi(y0)| − 2 ≤ |tr(σi(A))| ≤ 2 + 2|σi(y0)|. (7.3)

Proposition 7.3.4. Let I ⊂ Ok be an ideal with N(I) ≥ 40n2 and A ∈ Γ(I) with

y0 6= 0. Then for any point z = (z1, . . . , zn) ∈ (H2)n we have

d(H2)n(z, Az) ≥ 4√n

log(N(I))− 2√n log(40).

Proof. By Lemma 7.3.2, |tr(σj(A))| ≥ 8 for some j ∈ {1, . . . , n}, hence wecan subdivide our analysis into two different cases:

Case 1: |tr(σi(A))| ≥ 8 for any i = 1, . . . , n.

In this case all of the matrices σi(A) are hyperbolic and the right hand sideof Equation (7.3) implies that |σi(y0)| ≥ 3 for i = 1, . . . , n.

Using Equation (5.2.1), the left hand side of Equation (7.3), the fact that|σi(y0)| ≥ 3 for i = 1, . . . , n, the convexity of the function x2 and Lemma7.3.1 we obtain

87

d(H2)n(z, Az) =√d2H2(z1, σ1(A)z1) + · · ·+ d2

H2(zn, σn(A)zn))

≥ 2

√log2(|tr(σ1(A))| − 1) + · · ·+ log2(|tr(σn(A))| − 1)

≥ 2

√log2(2|σ1(y0)| − 3) + · · ·+ log2(2|σn(y0)| − 3)

≥ 2

√log2(|σ1(y0)|) + · · ·+ log2(|σn(y0)|)

≥ 2√n

(log(|σ1(y0)|) + · · ·+ log(|σn(y0)|))

=2√n

log(|N(y0)|)

≥ 4√n

log(N(I))− 2√n log(8).

Case 2: There are exactly k < n of the indices 1, . . . , n such that |tr(σj(A))| <8.Without loss of generality we assume that |tr(σj(A))| < 8 for j = 1, . . . , k.By the left hand side of Equation (7.3), |σj(y0)| < 5 for any of these j′s andby Lemma 7.3.1 we have

n∏i=k+1

|σi(y0)| = |N(y0)|∏ki=1 |σi(y0)|

>1

5n.8nN(I)2.

Now, as |tr(σi(A))| ≥ 8 for i = k + 1, . . . , n, for these indices σi(A) is hy-perbolic and |σi(y0)| ≥ 3 by the left hand side of Equation (7.3). By usingEquation (5.2.1) and the previous facts we obtain

88

d(H2)n(z, Az) =√d2H2(z1, σ1(A)z1) + · · ·+ d2

H2(zn, σn(A)zn))

≥√d2H2(zk+1, σk+1(A)zk+1) + · · ·+ d2

H2(zn, σn(A)zn))

≥ 2

√log2(|tr(σk+1(A))| − 1) + · · ·+ log2(|tr(σn(A))| − 1)

≥ 2

√log2(2|σk+1(y0)| − 3) + · · ·+ log2(2|σn(y0)| − 3)

≥ 2

√log2(|σk+1(y0)|) + · · ·+ log2(|σn(y0)|)

≥ 2√n− k

(log(|σk+1(y0)|) + · · ·+ log(|σn(y0)|))

=2√n− k

log

(n∏

i=k+1

|σi(y0)|

)

≥ 4√n

log(N(I))− 2√n log(40).

In both cases we get

d(H2)n(z, Az) ≥ 4√n

log(N(I))− 2√n log(40).

Corollary 7.3.5. For any ideal I ⊂ Ok with N(I) ≥ 40n2 , the length of any

non-contractible closed geodesic α in MI satisfies

`(α) ≥ 4√n

log(N(I))− 2√n log(40).

Proof. By Corollary 7.3.3, MI is a Riemannian manifold with the metric in-duced from (H2)n. If we lift α to a geodesic α = (α1, . . . , αn) in its universalcover (H2)n there is A ∈ Γ(I) acting on α as a translation and for any z inthe graph of α we have `(α) = d(H2)n((z, Az)). Since α is non-contractible,α is not a point, then for some i ∈ {1, . . . , n} the curve αi is a non trivialgeodesic in H2, and so σi(A) acts on it as a translation. This implies thatσi(A) is hyperbolic and, in particular, |tr(A)| 6= 2. Since |tr(A)| 6= 2 impliesy0 6= 0, the result now follows from Proposition 7.3.4.

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7.4 The index [Γ : Γ(I)]

In order to relate the systole to the index [Γ : Γ(I)] we need to estimatethis index in terms of N(I). In particular, we need to find uniform boundsfor the quotient [Γ:Γ(I)]

N(I)3, for ideals I ⊂ Ok with norm sufficiently large. The

result in this section can be deduced from general theory, but we prefer togive a concrete analysis that is relevant to our problem.

Lemma 7.4.1. For any ideal I ⊂ Ok

|SL2(Ok/I)| = N(I)3∏p|I

(1− 1

N(p)2

).

Proof. If I = pr11 . . . prss is the decomposition of the ideal I in prime ideals,then by the Chinese Remainder Theorem (Theorem 3.5.4)

SL2(Ok/I) ∼= SL2(Ok/pr11 )× · · · × SL2(Ok/prss ).

Therefore, the computation of |SL2(Ok/I)| reduces to the case I = pt wherep is a prime ideal and t > 0.

The quotient ring Ok/pt is a local ring with maximal ideal p/pt, |Ok/pt| =

N(p)t and |p/pt| = N(p)t−1. With the condition ad − bc = 1 for a, b, c, d ∈Ok/pt, we will compute |SL2(Ok/pt)| directly by separating the analysisinto two cases.

The first case holds when c ∈ p/pt. In this case neither a nor d are inp/pt hence they are units in Ok/pt. Therefore, since a, b and c determine dand there are N(p)t−1 possibilities for c, N(p)t possibilities for b and N(p)t−N(p)t−1 possibilities for a, we have N(p)3t−2(N(p)−1) matrices in SL2(Ok/pt)with c ∈ p/pt.

The other possibility is that c /∈ p/pt. In this case c is a unit and we haveN(p)t−1(N(p)− 1) possibilities for c; since a, c and d determine b and thereare N(p)t possibilities for a and d , we have N(p)3t−1(N(p)− 1) elements inSL2(Ok/pt) with c /∈ p/pt.

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In conclusion, |SL2(Ok/pt)| = N(p)3t(

1− 1N(p)2

).

Lemma 7.4.2. For all buy finitely many ideals I ⊂ Ok we have

ζk(2)−1 N(I)3 ≤ [Γ : Γ(I)] < N(I)3, (7.4)

where ζk denotes the Dedekind zeta function of k.

Proof. A well-known corollary of the strong approximation theorem (cf.[35, Theorem 7.15]) implies that for all but finitely many ideals I ⊂ Ok thereduction map

SL2(Ok)πI−→ SL2(Ok/I)

is surjective. For those ideals the index [Γ : Γ(I)] is equal to the cardinalityof SL2(Ok/I), which is given by the formula

N(I)3∏p|I

(1− 1

N(p)2

)by Lemma 7.4.1. From this the right-hand side of Inequality (7.4) followseasily. On the other hand, note that the product formula for the Dedekindzeta function of k (Section 3.5) says that

ζk(2) =∏p⊂Ok

1

1− N(p)−2≥∏p|I

1

1− N(p)−2.

This proves the second inequality.

7.5 The lower bound for sys1(MI)

Now we can present the proof of the main theorem of this chapter.

Theorem C. Let k be a totally real number field of degree n andOk be the ring ofintegers of k. Any sequence of ideals in Ok with N(I)→∞ eventually satisfies

sys1(MI) ≥4

3√n

log(vol(MI))− c,

91

where Γ(I) is the principal congruence subgroup of Γ = PSL2(Ok) at level I ,MI = Γ(I)\(H2)n and c is a constant independent of I .

Proof. For any ideal I with N(I) ≥ 40n2 , Corollary 7.3.3 implies that MI is

a Riemannian manifold with the metric induced by the product metric on(H2)n. Now, by Corollary 7.3.5 and Lemma 7.4.2, we conclude that

sys1(MI) ≥4

3√n

log([Γ : Γ(I)])− 2√n log(40)

when N(I)→∞.

7.6 Sharpness of the lower bound

To finish, we prove that among congruence coverings of Hilbert modularvarieties the constant 4

3√n

in the growth of the systole in general cannot beimproved to any γ > 4

3√n.

Theorem D. Let k be a totally real number field of degree n andOk be the ring ofintegers of k. Then there exists a sequence of ideals in Ok with N(I) → ∞ suchthat

sys1(MI) ≤4

3√n

log(vol(MI)) + c1,

where Γ(I) is the principal congruence subgroup of Γ = PSL2(Ok) at level I ,MI = Γ(I)\(H2)n and c1 is a constant independent of MI .

Proof. Let p be a rational integer and consider the ideal Ip = pOk in Ok.Since N(Ip) = pn, by following the same argument as in Section 7.2 withthe matrix

B =

(1− p2 p

−p 1

),

we obtain that sys1(MIp) ≤ 4√n log(p) when p is large enough. Therefore,

Lemma 7.4.2 implies that

sys1(MIp) ≤4

3√n

log([Γ : Γ(Ip)]) +4

3√n

log(ζk(2)),

92

when p → ∞, and then we obtain the result with c1 = 43√n

log(

ζk(2)vol(Mk)

)where Mk = Γ\(H2)n.

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FINAL CONSIDERATIONS

We now address with the work done so far and we discuss some questionswhich remain open and which can lead to a future research.

The natural question originating from the Buser-Sarnak construction iswhether there exists an n-manifoldM with systole greater than 4

3log(vol(M)).

The results in this thesis discard the possibility to find such a manifoldfrom congruence coverings of arithmetic manifolds of the first type, andcongruence coverings of Hilbert modular varieties, and in this sense thequestion remains open.

The first assumption in Theorem A is that the manifold M is an arithmetichyperbolic manifold of the first type. As it was pointed out previously, ineven dimensions all of the compact arithmetic hyperbolic manifolds are ofthe first type but in odd dimensions there is a second class which are con-structed from skew-hermitian forms over division quaternion algebras. Indimension 7, there is a third class constructed using certain Cayley alge-bras. A natural next step is to study congruence coverings in the secondclass. It is interesting that the constant appearing in the systolic boundcould be different from 8

n(n+1). Another interesting fact is that the geomet-

rical argument used in the proof of Theorem B only works in the first typesetting (cf. [29, Sec. 1]). Therefore, if a lower bound for the systole were tobe found in the second class, a different technique would be necessary in

94

order to prove its sharpness.

On the other side, in Theorem A the congruence subgroups which havebeen considered are associated with prime ideals. This assumption comesfrom the fact that orthogonal groups of quadratic forms over finite fieldshave been largely studied and there are well-established formulas of theirorders. We expect that Theorem A holds without this assumption.

In the Hilbert modular setting, we find that Theorem C holds for any ideal,and not only for the prime ones. We are able to do so due to the fact thatwe can find an explicit bound of the order of the group SL2 with entries ina finite ring. However, to prove the sharpness of the constant 4

3√n

in The-orem D we consider a special family of ideals. We believe that this resultalso holds for any ideal.

We finish these remarks by noting that congruence coverings of Hilbertmodular varieties and arithmetic hyperbolic manifolds of the first typeare the only types of arithmetic manifolds where the precise growth ofsystole of their congruence coverings is known so far. We hope that similarresults for the systole in other arithmetic locally symmetric spaces can beestablished in the future.

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