Principal congruence link complements

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Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

Principal congruence link

complements

Candidato:

Relatore:

Ludovico Battista

Prof. Bruno Martelli

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Introduction

The study of 3-manifolds plays a signicant role in low-dimensional topology. The proof of the Geometrization conjecture moved the attention to a specic class of them, the hyperbolic ones, and even if a complete characterization seems still dicult to obtain, a lot of work is spent in this direction.

We will focus on two particular families of hyperbolic 3-manifolds: the principal congruence manifolds and the link complements.

Link complements are dened in a purely geometric way: they are the complement of the image of a smooth embedding of a disjoint union of circles in the 3-dimensional sphere S3_.

We can dene principal congruence manifolds in an exquisitely algebraic way. Every complete connected orientable hyperbolic 3-manifold is isometric to a quotient H3_{/G, where G is a discrete subgroup of Isom}+

(H3). Hence, we may try to nd discrete subgroups of Isom+

(H3₎ _{to obtain hyperbolic}

3-manifolds.

We may identify the orientation-preserving isometry group of hyperbolic space with P SL(2, C), the group of 2 × 2 matrices with coecient in C and determinant 1 up to multiplication by scalars, and then it is natural to look for subrings R ⊆ C that give rise to discrete subgroups P SL(2, R) ⊆ P SL(2, C). We will dene some subrings with this property, and we will call them Od, where d is a square-free positive integer.

Given an ideal I of Od, we can dene the following subgroups of P SL(2, Od):

Γ(I) := Kern_{P SL(2, O}d) → P SL

2, Od_/I

o .

We call a hyperbolic manifold of the form H3_{/Γ(I) a principal congruence} manifold.

Baker, Goerner and Reid in [BRG18] classied the intersection of these two families, indeed they provided a complete nite list of hyperbolic 3-manifolds that are both principal congruence 3-manifolds and link comple-ments. In particular, they proved:

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Theorem 0.1. The following list of 48 pairs (d, I), where d is a natural number and I is an ideal of Od, describes all the subgroups Γ(I) < P SL(2, Od)

such that H3_{/Γ(I) is a link complement in S}3_:

1. d = 1: I = h2i, h2 ± ii, h(1 ± i)3_{i, h3i, h3 ± ii, h3 ± 2ii, h4 ± ii}_.

2. d = 2: I = h1 ±√−2i, h2i, h2 ±√−2i, h1 ± 2√−2i, h3 ±√−2i.

3. d = 3: I = h2i, h3i, h(5 ±_√ √−3)/2i, h3 ±√−3i, h(7 ±√−3)/2i, h4 ± −3i, h(9 ±√−3)/2i.

4. d = 5: I = h3, (1 ±√−5)i.

5. d = 7: I = h(1±_√ √−7)/2i, h2i, h(3±√−7)/2i, h±√−7i, h1±√−7i, h(−5± −7)/2i, h2 ±√−7i, h(7 ±√−7)/2i, h(1 ± 3√−7)/2i.

6. d = 11: I = h2, (1 ±√−11)/2i, h(3 ±√−11)/2i, h(5 ±√−11)/2i.

7. d = 15: I = h2, (1±_√ √−15)/2i, h3, (3±√−15)/2i, h4, (1±√−15)/2i, h5, (5± −15)/2i, h6, (−3 ±√−15)/2i.

8. d = 19: I = h(1 ±√−19)/2i.

9. d = 23: I = h2, (1 ±√−23)/2i, h3, (1 ±√−23)/2i, h4, (−3 ±√−23)/2i. 10. d = 31: I = h2, (1 ±√−31)/2i, h4, (1 ±√−31)/2i, h5, (3 ±√−31)/2i. 11. d = 47: I = h2, (1 ±√−47)/2i, h3, (1 ±√−47)/2i, h4, (1 ±√−47)/2i. 12. d = 71: I = h2, (1 ±√−71)i.

The elegance of this theorem descends from the fact that, as we saw, these two families are dened in very dierent ways. Theorem 0.1 is the result of the eorts of several mathematicians: Vogtmann in [Vog85] stated the following theorem:

Theorem 0.2. Let I be an ideal of Odsuch that H 3

/Γ(I) is a link complement. Then

d ∈ {1, 2, 3, 5, 6, 7, 11, 15, 19, 23, 31, 39, 47, 71}.

Baker and Reid in [BR14] proved that, for each value of d, there are only nitely many ideals I ⊂ Od such that H

3

/Γ(I) can be a link complement; this implies in fact that there are only nitely many principal congruence manifolds that are link complements. In the same article, they started to classify the pairs (d, I) that describe such manifolds, handling some particular

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cases of the Theorem 0.1; in [BR17], they completed the classication for d = 5, 15, 23, 31, 47, 71.

Goerner in [Goe11] constructed two link complements dieomorphic to the manifolds H3_{/Γ(I) in the cases d = 3 and I = h3i, h3 +} √−3i, and in [Goe15] completed the classication in the cases d = 1, 3.

The thesis is structured as follows: in the rst chapter we recall the basic notions of hyperbolic geometry and we give a specic attention to the presentation of arithmetic manifolds.

In the second chapter we introduce the rst results that allow us to study whether a principal congruence manifold is indeed a link complement in S3_:

Vogtmann in 1985 and Baker in 2001 listed, using cuspidal cohomology, a sequence of all values of d that can produce a principal congruence manifold dieomorphic to a link complement. On the other hand, thanks to a work of Adam and Reid in 2000, a bound on the minimal lenght of closed geodesics in such manifolds allow us to control the norm of the ideal I. The chapter ends with the presentation of a constructive method, exposed by Baker and Reid in 2014, that permits to verify if a manifold M = H3_{/Γ(I) is indeed a} link complement.

In the third chapter we introduce, following a presentation of Goerner in 2014, the study of the manifolds that can be decomposed by regular tessel-lations. This construction is linked to principal congruence manifolds when d = 1 and d = 3. Thanks to this connection, it is possible to nd a more accurate bound on the norm of the ideal I and to implement an algorithm that allows us to manipulate these manifolds from a combinatoric point of view.

In the fourth and last chapter we present an example of a construction of a link whose complement is dieomorphic to H3_{/Γ(h3i) when d = 3, described} in Goerner's Ph.D. thesis in 2011.

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Chapter 1 Preliminaries

This chapter is a recap about (arithmetic) hyperbolic manifolds. Its aim is to give denitions and results that are needed in the rest of the thesis. Also, we want to give references in order to allow the reader to deepen this argument.

1.1 Hyperbolic 3-Manifolds

A hyperbolic 3-manifold M is a connected complete Riemannian manifold locally isometric to the hyperbolic space H3_.

There are several ways to interpret these objects, but we are specically interested in visualizing them as a quotient of H3 _{by a subgroup of}

isome-tries: let M be a hyperbolic 3-manifold; its universal cover is a hyperbolic, connected, complete and simply connected 3-manifold. It is possible to prove that these properties characterize the hyperbolic space, and therefore M is homeomorphic to H3_{/Γ, where the action of Γ} ∼_{= π}

1(M ) on H3 is free and

properly discontinuous. A priori, Γ is just contained in Omeo(H3₎_{, but,}

acting as deck transformations, its elements are local isometries, and then isometries.

The group Isom(H3₎ _{is isomorphic to P SL(2, C): take an element A ∈}

P SL(2, C) and let it act on the boundary of H3 as a Möbius transformation; it is possible to extend this action on the whole hyperbolic space; the resulting homeomorphism is in fact an isometry.

Provided with this topology, it is possible to prove that a subgroup Γ < Isom(H3_{) ∼}

= P SL(2, C) acts properly discontinuously on H3 if and only if Γ is discrete, i.e. the subspace topology is discrete. In this setting, acting freely is equivalent to be torsion-free.

Theorem 1.1. If M is a hyperbolic 3-manifold, then M is isometric to H3_/Γ, where Γ is a discrete and torsion-free subgroup of P SL(2, C).

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It is then natural to study the discrete subgroups of P SL(2, C), which are called Kleinian groups, and, among them, the torsion-free ones.

By P SL(2, C) we mean the group of invertible 2 × 2 matrices with coef-cients in C up to multiplication by non-zero elements in C. By P SL(2, R), where R is a ring, we mean SL(2, R) up to multiplication by ±1.

1.1.1 Kleinian subgroups

It is possible to classify the elements in P SL(2, C) depending on their xed points on H3 _{and its boundary; let γ be an element of P SL(2, C), then:}

γ is elliptic if it has a xed point in H3_{; otherwise}

γ is parabolic if it has only one xed point in ∂H3_;

γ is hyperbolic if it has two xed points in ∂H3_.

This denition brings some considerations: every isometry of H3 _{has at least}

one xed point in H3, because its extension is continuous and we can apply

the Brower theorem. Furthermore, it is impossible to x three or more points in the boundary without xing points in H3_{. These geometric properties can}

be read also through the trace of the element γ: γ is elliptic if tr(γ) ∈ R and |tr(γ)| < 2; γ is parabolic if tr(γ) ∈ {−2, 2};

γ is hyperbolic otherwise.

If γ is parabolic, then it is conjugated to the element z 7→ z + 1, where the xed point is ∞. In all other cases, there is a geodesic, often called axis of γ, that is xed as a set by γ: if γ is elliptic, then γ is a rotation along that axis; if γ is hyperbolic, it is a rototranslation along it.

If Γ is a Kleinian group and ζ ∈ ˆC ∼= ∂H3, we denote by Γζ the set of

elements in Γ which x ζ. After conjugation, if necessary, we can suppose that ζ is ∞ in the upper space model, and that Γζ is a subgroup of B, the

stabilizer of ∞ in P SL(2, C), which is B =a b 0 a−1 |a ∈ C∗, b ∈ C .

It is possible to characterize the discrete subgroups of B, which fall in the following classes:

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A nite extension of an innite cyclic group generated by a hyperbolic or a parabolic element;

A nite extension of Z ⊕ Z, which is generated by a pair of parabolics. The last case is of very special interest:

Denition 1.2. A point ζ ∈ ˆC is a (rank-2) cusp of a Kleinian group Γ if Γζ contains Z ⊕ Z as a subgroup.

Observation 1.3. If Γ is torsion free, the stabilizer of a cusp is necessarily Z ⊕ Z.

During this thesis we will always deal with a special class of Kleinian groups, that we now dene.

Denition 1.4. A Kleinian group Γ is

of nite covolume if the manifold H3_{/Γ has nite volume;} cocompact if the manifold H3_{/Γ is compact.}

The principal congruence groups we will be interested in are always of nite covolume and never cocompact.

Observation 1.5. Even if we are mostly interested in torsion-free subgroups, that give rise to manifolds, we will often study properties of Kleinian groups without this hypothesis. The topological space obtained as a quotient H3_/Γ, where Γ is not torsion-free, is an orbifold; these objects can be provided with some geometric structure: for more information see the end of the chapter.

We study orbifolds because they have relations with manifolds; the rst one that we can easily dene is the following:

Denition 1.6. Let Γ1 and Γ2 be Kleinian groups. We say that they are

commensurable if Γ1∩ Γ2 is of nite index in both Γ1 and Γ2;

commensurable in the wide sense if Γ1 and a conjugate of Γ2 are

com-mensurable.

Obviously, an equivalence relation is interesting if it is possible to asso-ciate invariants to its elements. For example, the set of cusp points in ˆC is invariant under commensurability, and the same holds for being of nite covolume.

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1.1.2 Thick-thin decomposition

There are several convenient ways to decompose a hyperbolic manifold M. We begin with a denition:

Denition 1.7. Let X be a Hausdor locally compact space. For compact K ⊂ X dene

E(K) = {E | E is a connected component of X r K, E is not compact}. If there exists a compact set K ⊂ X such that for any other compact set K0 ⊇ K the inclusion induces a bijective mapping E(K0_{) → E (K)}_{, then we}

shall call each element of E(K) a topological end (or simply an end) of X relative to K.

In order to make this denition reasonable, we need to remark that if H is another compact with the same property, then there exists a natural bijection E(K) → E(H). The idea is that the ends of a manifold are "the parts that go to innity".

Recall the denition of injectivity radius of M:

Denition 1.8. Let M be a complete Riemannian manifold; the injectivity radius injp(M )of M at a point p is the supremum of all r > 0 such that the

exponential map is dened on the ball centered in 0 of radius r in TpM and

expp|B(0,r) is a dieomorphism onto its image.

Let now M be a hyperbolic manifold and > 0 be a constant. Denition 1.9. The thick part of M, denoted by M[,∞), is the subset

M[,∞) = n x ∈ M | inj_x(M ) ≥ 2 o .

The thin part is dened as the closure of the complement: M(0,] = M r M[,∞)

If M has nite volume, then the thick part is compact; we can then use n

M_[1 n,∞)

o

n∈N for the denition of topological ends. This is a useful way to

interpret the thin part: it (denitely) consists of the part of M that goes to innity.

In the 3-dimensional case, much more can be obtained:

Theorem 1.10. Let M = H3_{/Γ be a nite-volume orientable hyperbolic} man-ifold. Then:

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1. M is dieomorphic to the interior of a compact manifold N, whose boundary consists of tori;

2. for every boundary component T of ∂N, the homomorphism i∗ : π1(T ) → π1(N ) ∼= π1(M )

induced by inclusion is injective, and its image (read in Isom+

(H3₎)

consists of parabolic elements xing the same point on ∂H3_;

3. The xed point of every parabolic element is a cusp;

4. The function in 2 induces a bijection between the set of orbits of cusps of Γ under the action of Γ and the boundary components of N.

The theorem contains all the information we need about cusps. First of all, often the name cusp has a dierent meaning: the quotient of H3 _{by a}

discrete subgroup generated by parabolics xing ∞ acting freely. We will call these topological cusps. If M is nite-volume and is small enough, the thin part of M consists of truncated topological cusps dieomorphic to T2_{×[0, ∞)}_,

where T2 _{is the torus. Retracting these topological cusps, it is possible to}

nd the dieomorphism in Theorem 1.10 - (1). If M is not compact, Γ contains at least a parabolic, and the stabilizer of this xed point necessarily contains another parabolic element, generating a Z ⊕ Z altogether. Every Γ-equivalence classes of cusps determines a topological cusp in the thin part of M.

Proposition 1.11. If Γ is a torsion-free Kleinian group of nite covolume, then Γ is cocompact if and only if it does not contain any parabolic element. Proof. If Γ contains a parabolic element γ, then we can conjugate so that γ acts like the translation (z, t) 7→ (z + 1, t) in the upper space model C × R+_.

If we consider the elements (0, n), the injectivity radius of their images tends to zero as n goes to innity, then injx(M ) = 0 and M cannot be compact.

The converse part is an easy consequence of Theorem 1.10.

1.1.3 Systoles

We dene a closed geodesic in a hyperbolic 3-manifold M to be a smooth map α : S1 _{→ M} _{such that its lifting ˜α : R → H}3 _{is a geodesic. There exists}

a bijection

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Elements in the same conjugacy class [A] in P SL(2, C) have the same trace, which we will denote by tr([A]), dened up to sign. From this it is possible to deduce the length of the associated closed geodesic:

Lemma 1.12. Let α be a closed geodesic corresponding to a conjugacy class [A] ⊂ P SL(2, C). Then

cosh(`(α)/2) ≥ |tr([A])|/2.

Proof. We can suppose that the element A in [A] xes 0 and ∞; hence A is of the form

A =λ 0 0 λ−1

.

Let p = (0, 1) ∈ C × R+ _{be a point in the xed line l for A. The length}

of the closed geodesic α is the distance between p and Ap. Since the element

B =arg(λ

−1₎ ₀

0 arg(λ)

xes all the points of l, we can consider the image of p under the composition AB; notice that

AB =|λ| 0 0 |λ−1_|

.

The element ABp is equal to (0, |λ|2₎_{, hence the distance between p and}

ABp is equal to: ˆ |λ|2 1 1 zdz = − ln(|λ 2_{|) = −2 ln(|λ|) = `(α).} Then cosh `(α) 2 = cosh(− ln(|λ|)) = |λ| + |λ −1_| 2 ≥ |λ + λ−1_| 2 = |tr(γ)| 2 .

It is possible to prove that there exist closed geodesics of minimal length, which we will call systoles. We will denote by sl(M) the length of a systole in M.

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1.1.4 (Hyperbolic) Dehn lling

Let M be a 3-manifold with boundary. Suppose that a component T of the boundary is homeomorphic to the torus. Then it is possible to "ll" this component, in several ways:

Denition 1.13. A Dehn lling of M is the result of gluing a solid torus D2_{× S}1 _{to T via a dieomorphism ∂D}2_{× S}1 _{→ T}_.

We are interested on what happens to the topology of M and to its geometry after this operation. First of all, we note that the closed curve ∂D2 _{× {1}} _{is identied with some closed curve τ ⊂ T ; this is the only}

information we need:

Fact 1.14. The dieomorphism class of the Dehn lling of M depends only on the unoriented isotopy class of τ.

It is a theorem that the unoriented isotopy classes of closed curves on the torus are parametrized by Q ∪ {∞}. In particular, x two generators µ and λ of π1(T ): we think of them as a meridian and a longitude, respectively.

Given a pair of coprime integers (p, q), it is possible to nd a closed curve whose homotopy class is pµ + qλ, and every closed curve is isotopic to such a curve (if p or q is zero, we require the other one to be one). We then associate

(p, q) 7→ p q, where we dene 1

0 to be ∞. One proposition that will have really important

consequences is the following (the notations are the same as before): Proposition 1.15. Let M1 be a Dehn lling of M. Then

π1(M1) = π1(M )/_{N (τ ),}

where N(τ) is the normal closure of {τ}, that is the smallest normal group that contains it.

Proposition 1.15 is the reason why we say that such a Dehn lling kills τ. Clearly, the proposition extends trivially to the case we operate more Dehn llings on distinct boundary tori: in that case the resulting fundamental group is the one of M modulo the normalizer of the union of the closed curves that we identied with the ∂D × S1_.

One of the most interesting results about hyperbolic 3-manifolds regards the geometry of these llings; we state it here, even if we will not need it.

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Theorem 1.16 (Hyperbolic Dehn lling). Let M = Int(N) be a complete orientable nite-volume hyperbolic three-manifold where ∂N = T1 t . . . t Tc.

For every i = 1, . . . , c there is a nite set Si of isotopy classes of closed

curves in Ti such that for every Dehn lling on curves τ1, . . . , τc with τi ∈ S/ i

for all i, the lled manifold is a complete orientable nite-volume hyperbolic manifold.

This theorem tells us that most of the llings of a hyperbolic manifold are indeed hyperbolic; then the "interesting" Dehn llings turn out to be the ones that produce a non-hyperbolic manifold: these llings are called exceptional Dehn llings.

Observation 1.17. There is a very simple case where at least one excep-tional Dehn lling is easy to detect: if L is a link in S3_{and M is dieomorphic}

to S3

r L, then the trivial Dehn lling (the one that produces S3) is an ex-ceptional Dehn lling.

This Dehn lling is exceptional because S3 _{does not admit any hyperbolic}

structure, the easiest way to see this is that its universal cover (being S3_{) is}

not dieomorphic to R3.

1.2 Some algebraic number theory

During the thesis we will need some basic results about algebraic number theory; we will rapidly summarise them in this section. To study in deep these arguments, see [Mar77] or [KI90].

1.2.1 Number elds

Take an element α ∈ C; we say that α is algebraic over Q (or simply alge-braic) if there is a polynomial p ∈ Q[x] such that p(α) = 0.

A number eld K ⊃ Q is a eld that contains Q and has nite dimension as a Q-vector space. There are a lot of ways to obtain such an object; if {αi}_i=1,...,k is a nite set of algebraic elements, then

Q[α1, . . . , αk] :=

\

Lelds s.t. {αi}⊂L⊂C

L

is a number eld. It is a theorem that every such eld can be obtained extending Q with only one element:

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Theorem 1.18 (Primitive element). If L is a number eld, there exists t ∈ L such that

L = Q[t].

In this setting, we can consider a monomorphism ϕ from L to C; since the restriction ϕ||Q is necessarily the identity, the map ϕ depends only on

the value ϕ(t). Let µt(x) be the minimal polynomial of t over Q, then

µt(x) = c

Y

i=1

(x − ti),

where ti are the roots of µt. Then ϕ(t) ∈ {ti}_i=1,...,c. We have then c possible

monomorphism from L to C. If ϕ(t) is not real, then ϕ (the monomorphism obtained by postcomposing with conjugation in C) is another monomor-phism.

Denition 1.19. Given L = Q[t], we dene a place to be: the monomorphism ϕ, if ϕ(t) ∈ R;

the set {ϕ, ϕ} if ϕ(t) /∈ R.

If r1 is the number of real places (the ones of the rst type) and r2 is the

number of complex ones, then

c = dim_Q(L) = r1+ 2r2.

1.2.2 Ring of integers

Let L be a number eld.

Denition 1.20. An element α ∈ L is an algebraic integer if it is the root of a monic polynomial with coecients in Z.

The denition of algebraic integers is a particular case of the following: Denition 1.21. Let R be a subring of the commutative ring A. Then

α ∈ A is said to be an integer over R if it satises a monic polynomial with coecients in R;

In the same setting as above, the set of all elements of A that are integers over R is dened to be the integral closure of R in A;

If the integral closure of R in A is R itself, then R is said to be integrally closed in A.

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The set of algebraic integers in L is the integral closure of Z in L. It is possible to prove that this is a subring of L. We will denote it by RL. It is

possible to prove that RL always admits a Z − basis: such a set is called an

integral basis of L.

Integral bases help us to dene a strong invariant of a number eld. Let {α1, . . . , αn} be an integral basis of RL; the discriminant of such a basis is

dened as:

∆L= det(ϕi(αj))2_i,j=1,...,n.

Where the ϕi are the possible monomorphism from L to C. Every integral

bases gives rise to the same value ∆L, called the discriminant of L.

Ring of integers of number elds have several properties that we will need. Let's start with an independent denition:

Denition 1.22. Let D be an integral domain with elds of fractions K. Then D is a Dedekind domain if all the following conditions hold:

1. D is noetherian;

2. D is integrally closed in K;

3. Every non-zero prime ideal in D is maximal (that is, its Krull dimension is one).

The ring of integers in a number eld is always a Dedekind domain. Theorem 1.23. Let RL be the ring of integers of a number eld L. Then

RL is a Dedekind domain and its eld of fraction is L;

if I is a non-zero ideal of RL, the quotient RL/I is a nite ring;

if I is an ideal of RL and a is a non-zero element in I, then there exists

a b ∈ I such that I = ha, bi.

There are several theorems that allow us to understand more deeply the structure of Dedekind domains.

We are specically interested in ideals in Dedekind domains; rstly, we dene a generalization of ideals:

Denition 1.24. Let D be a Dedekind domain with eld of fractions K. Then a D-submodule A of K is a fractional ideal if there exists some a ∈ D such that aA ⊂ D.

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Clearly, ideals in D are fractional ideals. It is possible to give a group structure to the set of fractional ideals, where multiplication is the multi-plication between sets, the identity element is D and the inverse is given by

A−1 = {x ∈ K|xA ⊂ D}.

This group will have a strong importance in our study, because it is connected with the geometry of principal congruence manifolds.

Anyway, for the moment we use it to describe the structure of ideals of D:

Theorem 1.25. Let D be a Dedekind domain; then Every non-zero ideal I of D is of the form

I = Pa1 1 P a2 2 . . . P ar r

where Pi are prime ideals of RL. This decomposition is unique (up to

permutations).

The set of fractional ideals of D is a free abelian group freely generated by the set of all the prime ideals in RL.

In virtue of Theorem 1.23, it is possible to associate a natural number to every non-zero ideal I; we dene the norm of I to be

N (I) = RL_/I .

The norm function is completely multiplicative: N(IJ) = N(I)N(J). In the case of number elds and principal ideals, the norm has a really simple representation; for an element x ∈ RL, we dene the norm of x over Q to be

N_L|Q(x) =

n

Y

i=1

ϕi(x)

where ϕiare the monomorphism from L to C. Then for every nonzero element

x ∈ RL

N (xRL) = NL|Q(x).

We will need the following result about norms of ideals:

Lemma 1.26. Let RL be the ring of integers of a number eld. Then the set

nm = {I ⊂ RL ideals |N(I) < m}

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Proof. Let's divide the proof in several steps:

1. Given a prime ideal P in RL, the quotient RL/P has cardinality pc for

a natural prime p ∈ N and a natural c ∈ N.

That's a consequence of the fact that RL_{/P is nite (Theorem 1.23) and}

that it is a eld (because every prime ideal is maximal). Then RL_{/P is}

a nite eld and so is Fpc for some p, c ∈ N.

2. Fix a prime η ∈ N. There are only nitely many prime ideals P in RL

such that RL_{/P has cardinality η}c, with c ∈ N.

If P has cardinality ηc, then η ∈ P because the element 1 has order η

in the quotient (because in Fηc every non-trivial element has order η),

then η · 1 = η ∈ P.

Then prime ideals of this type need to contain the ideal hηi, hence P corresponds to one ideal in RL_{/hηi, that is a nite ring and then has a}

nite number of ideals.

3. The set of ideals with a given norm k is nite. Let k = pa1

1 p a2

2 . . . pass be the prime decomposition of k. If N(I) = k,

then N (I) = N (Pb1 1 P b2 2 . . . P br r ) = N (P1)b1. . . N (Pr)br = pa11p a2 2 . . . p as s .

For every pi we have only a nite number of prime ideals we can choose,

and for any of these choices there only nite values of bi allowed.

The proof is complete.

The last denition we need is the one of class group. Let IL be the ideal

group of L, i.e. the group of fractional ideals of L. Let PL be the subgroup

of IL that consists of all the non-zero principal fractional ideals, the ones of

the form αRL for some α ∈ L∗. Then:

Denition 1.27.

The class group of L, denoted by CL, is the quotient IL/P_L.

The class number of L is the cardinality of the class group.

It can be proved that the class number of a number eld is always nite. It measures the ratio between all fractional ideals and principal ones; RL is

a PID if and only if its class number is 1. We should remark that calculating the class number is dicult: even in the case of quadratic extensions, the values of class numbers are not known.

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Observation 1.28. Choose an equivalence class [A] in the class group. The fractional ideal A can be interpreted as a product

A = Pa1

1 . . . P as

s

where as ∈ Z. For every Pi we are able to nd a principal ideal hxii such

that

hxii = Pibi· · · . . . with bi > 0,

then we can construct the "fractional" ideal B = Ahx1i

|a1|

. . . hxsi |as|

;

hence we will be able to nd a representative of [A] in the class group that is an ideal in RL.

1.3 Arithmetic hyperbolic 3-manifolds

There is a really special class of hyperbolic 3-manifolds, the arithmetic ones. Thurston posed several questions in [Thu82], the nineteenth one being "Find topological and geometric properties of quotient spaces of arithmetic subgroups of P SL(2, C). These manifolds often seem to have special beauty". We need some algebraic tools to dene them; this section is mostly motivational, it can be skipped if necessary.

1.3.1 Quaternion algebras

We will only need some basic denition. Let F be a eld of characteristic 6= 2.

Denition 1.29. A quaternion algebra over F is a four dimensional F -vector space with basis {1, i, j, k}, where multiplication is dened by requiring that 1 be a multiplicative identity, that

i2 = a1; j2 = b1 ij = −ji = k

for some a and b in F∗_{, and by extending such multiplication linearly so that}

A is an associative algebra over F . As a consequence, k2 _{= −ab}.

One thing we will need is the extension of scalars:

Observation 1.30. If A is a quaternion algebra over F and ϕ is a monomor-phism from F to E, then

A ⊗F E

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There are some basic examples of quaternion algebras:

Example 1. 1. the Hamilton quaternions H form the most famous one; 2. for every eld F , the group of matrices M2(F ) is also a quaternion

algebra, the elements of the basis being 1 = 1 0 0 1 , i =1 0 0 −1 , j =0 1 1 0 , k = 0 1 −1 0 , and a = b = 1.

Over R, there are no other examples:

Fact 1.31. If A is a quaternion algebra over R, then A is isomorphic either to H or to M2(R);

If A is a quaternion algebra over C, then A is isomorphic to M2(C).

The fact that the Hamilton quaternions are a division algebra is not a uke:

Fact 1.32. A quaternion algebra is either a division algebra or it is isomor-phic to M2(F ).

If A is a quaternion algebra over a number eld L and {ϕi}_i=1,...,c is the

set of real places, where ϕi : L → R, then we say that:

Denition 1.33. A splits at ϕi if A ⊗LR is isomorphic to M2(R);

A is ramied at ϕi if A ⊗LR is isomorphic to H.

Trace and norm can be dened in every quaternion algebra, and these denitions coincide with the usual ones in the case of Example 1 - 2:

Denition 1.34. Let x = r + is + jt + ku be an element in a quaternion algebra A. Then

the reduced norm of x is dened as:

n(x) = (r + is + jt + ku)(r − is − jt − ku) = r2− s2_{a − t}2_{b + u}2_ab;

the reduced trace of x is dened as:

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1.3.2 Orders

We now introduce some objects in quaternion algebras that will help us to nd discrete subgroups of P SL(2, C): the orders. The study of these objects is one of the crucial tools needed to obtain results about arithmetic manifolds; nevertheless, here we just give their denition, and recommend the interested reader to consult the references at the end of this Chapter to obtain an insight on this topic.

Even if it is possible to give a denition in more general settings, here we will just handle the case where L is a number eld and RL is its ring of

integers.

Denition 1.35. If V is a L-vector space, an RL-lattice W in V is a nitely

generated RL-module contained in V . Furthermore W is a complete RL

-lattice if W ⊗RLL = V.

Denition 1.36. An order O is a complete RL -lattice which is also a ring

with 1.

Denition 1.37. We write O1 _{to indicate the set}

O1 = {x ∈ O| n(x) = 1}, where n is the reduced norm.

The only example we will need in our case is the following: if V = M2(L)

is a quaternion algebra over L, then O = M2(RL) is an order in it and

O1 _{= SL(2, R}

L) is the corresponding subgroup of elements with reduced

norm 1. That's because the reduced norm coincide with the usual norm in the matrix space, and therefore M2(RL)1 = SL(2, RL).

In some peculiar cases, the elements of norm 1 in orders embed in SL(2, C), and the image is (in the quotient P SL(2, C)) discrete and of nite covolume. We now give an idea of the construction of such embedding.

Let A be a quaternion algebra over the number eld L and RLbe its ring

of integers; let ϕ1, . . . , ϕc be the monomorphisms from L to C, where the

rst r1 are contained in R and the last 2r2 = c − r1 are not. Then

Lemma 1.38. If A is ramied at s1 real places, then

A ⊗_Q_R∼= s1H ⊕ (r1− s1)M2(R) ⊕ r2M2(C)

Idea. For every embedding ϕi, it is possible to nd a homomorphism

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where Bi is H if A ramies at ϕi and ϕi is real, M2(R) if it does not ramies

but is real, M2(C) otherwise (recall Fact 1.31). It is then possible to dene

A ⊗_Q_{R →}

c

M

i=1

Bi.

Pair of conjugate embeddings ϕi, ϕi+1 = ϕi in M2(C) have image contained

in the set

∆(M2(C)) = {(A, A) ∈ M2(C) ⊕ M2(C)} ∼= M2(C),

then it is natural to consider only one embedding for each complex place. This restriction is a isomorphism.

The composition of the natural embedding A → A⊗QR, this isomorphism

and the projection on the terms of the form M2(K), can be proved to be an

embedding

ψ : A → (r1− s1)M2(R) ⊕ r2M2(C) =: G.

We now have all the tools we need to state the main theorem:

Theorem 1.39. Let O be an order in a quaternion algebra A with at least one place where A ramies;

under the embedding ψ described above, the image ψ(O1₎ _{is discrete in}

G1 = (r1− s1) SL(2, R) ⊕ r2SL(2, C);

if G0 _{is a factor of G}1 _{with 1 6= G}0 _{6= G}1_{, then the projection of ψ(O}1₎

in G0 _{is dense in G}0_;

every other embedding of A in G diers from an inner automorphism of G;

if r1 − s1 = 0 and r2 = 1, then O1 embeds in SL(2, C); its image in

P SL(2, C) is a Kleinian group of nite covolume.

This theorem is the reason why we dene arithmetic groups in the fol-lowing way:

Denition 1.40. Let L be a number eld with exactly one complex place and let A be a quaternion algebra over L which is ramied at all real places. Let ρ be a L-embedding of A into M2(C) and let O be an RL-order of A.

Then a subgroup Γ of SL(2, C) (or P SL(2, C)) is an arithmetic Kleinian group if it is commensurable with some such ρ(O1₎_{(or Pρ(O}1₎_{). Hyperbolic}

3-manifolds and 3-orbifolds H3_{/Γ, will be referred to as arithmetic when their} covering groups Γ are arithmetic Kleinian groups.

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The Theorem 1.39 allows us to make some considerations that help us in accepting this denition:

Observation 1.41. If A has more than one complex place or does not ramify at a real place, the projection of O1 _{would be dense, and could}

not help us in nding Kleinian subgroups;

if O1 and O2 are two dierent orders in A, then O1∩O2 is also an order.

All of them being of nite covolume implies that the corresponding Kleinian groups are commensurable; the choice of the order O in the denition can then be random.

Dening these objects required a lot of work, and the theorems we stated are deeper than they can seem. Anyway, we will only need the fact that some specic subgroups are of nite covolume. The theorem we now state is the reason why we called this section motivational: it tells us that arithmetic subgroups that can be link complements need to be strictly related with the ones we are going to study.

Theorem 1.42. Let Γ be an arithmetic Kleinian group commensurable with Pρ(O1), where O is an order in a quaternion algebra A over L. The following are equivalent:

1. Γ is non-cocompact;

2. L = Q(√−d) and A = M2(L).

Proof. 1 ⇒ 2) If Γ is non ccoompact, then so is Pρ(O1₎_{, and then}

(Proposition 1.11) it contains a parabolic element. A division algebra cannot contain a parabolic element γ, because γ − Id is nilpotent, and then it is not invertible: just conjugate so that the xed point of γ is ∞ and it is clear. Hence A cannot be a division algebra; then (recall Fact 1.32) A ∼= M2(L).

If [L : Q] ≥ 3, then there should be a place where A ramies (because we allow only one complex place and all real places needs to ramify). But if A ∼= M2(L), then A⊗ϕR is naturally isomorphic to M2(R); then

[L : Q] < 3, but we want one complex place to exist, then L = Q(√−d). 2 ⇒ 1) In virtue of Observation 1.41, we can choose a random order in A to study Pρ(O1₎_{: if one of these is non-cocompact, the other ones}

also are, being commensurable with it; one way to prove this is using Proposition 1.11: if Γ has nite index in Γ0 _{and Γ}0 _{contains a parabolic}

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will be SL(2, R_Q(√

−d)). This subgroup contains parabolics, then the

quotient cannot be cocompact (Proposition 1.11).

1.4 Bianchi groups

Let S be a discrete subring of C; then P SL(2, S) ⊂ P SL(2, C) is a Kleinian group. There is a simple way to produce such rings: taking the ring of integers of a quadratic imaginary eld. Let d be a square-free positive integer, then

Notation. We denote by Kd the number eld Q(

√ −d); Od the ring of integers R_Q(√−d).

Some information about such rings of integers can be summarized very well due to the following facts, that hold for every a ∈ Z:

Fact 1.43. Let Q(√a) be a quadratic extension of Q and R_Q(√

a) be its ring

of integers. Then the following facts hold: RQ( √ a) = 1Z ⊕ waZ where wa = (√ a, if a ≡ 2, 3 mod 4 1+√a 2 , if a ≡ 1 mod 4 . The discriminant is ∆_Q(√ a)= ( 4a, if a ≡ 2, 3 mod 4 a, if a ≡ 1 mod 4 .

Denition 1.44. The Bianchi groups are the discrete subgroups of P SL(2, C) of the form P SL(2, Od). We denote them by Γd.

Clearly, Bianchi groups are discrete; it is possible to prove that they are of nite covolume by direct calculations. Anyway, they are exactly the Kleinian subgroups we obtained in the proof of Theorem 1.42; hence, being arithmetic Kleinian groups, they are discrete and of nite covolume.

Observation 1.45. If d is a positive square-free integer, the ring of integers of Q(√d) is not discrete. This is a consequence of Theorem 1.39, but can also be veried by hand.

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We are now going to study the topology of Bianchi orbifolds, i.e. the quotients H3_/Γ_d. The following result is a nice example of how algebraic number theory and geometric topology can be related:

Theorem 1.46. Let Γd= P SL(2, Od).

The cusp set of Γdis C(Γd) = P1(Kd) ⊂ P1(C), where P1(C) is identied

with the boundary of H3_.

The set of ends of H3_/Γ_d, i.e. P1(Kd)/

Γd, has cardinality hd, the class

number of Od (see Denition 1.27).

Proof. The identication between P1_{(C) and the usual C ∪ {∞} is}

[x, y] 7→ x y. Since [x, y] ∈ P1_(K

d)if and only if x_y ∈ Kd∪ {∞}, we will use this set

to prove the rst part.

The xed point of a parabolic element a b

c 2 − a

, where a, b, c ∈ Kd and 2a − a2− bc = 1, is either ∞ or:

az + b cz + 2 − a = z ⇒ az + b = cz 2_{+ 2z − az ⇒} ⇒ z = a − 1 ± √ 1 + a2_{− 2a + bc} c = a − 1 c ∈ Kd.

Conversely, given a point in z ∈ Kd, there exist α, β ∈ Od such that

z = α_β (recall that Kd is the eld of fractions of Od). Then z is xed by

1 + αβ −α2

β2 1 − αβ

∈ P SL(2, Od).

For the second part, we are going to dene a bijection between P1(Kd)/

Γd

and C, the class group of Kd (see Denition 1.27). The mapping

ϕ : P1(Kd) → C

dened by

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is well dened: if [x, y] = [f, g], then x = λf and y = λg, with λ ∈ Kd.

Then the corresponding fractional ideals diers by a principal ideal, λOd.

Since every ideal in Od is generated by two elements (Theorem 1.23),

and every class in the class group C admits a representative that is an ideal in Od (Observation 1.28), then ϕ is onto.

We now want to show that for [x, y], [x0_{, y}0

] ∈ P1_(K

d), there exist γ ∈ Γd

such that γ[x, y] = [x0_{, y}0_]_{if and only if [xO}

d+ yOd] = [x0Od+ y0Od]: If a b c d [x, y] = [x0, y0],

then there is a α ∈ Kdsuch that ac + by = αx0 and cx + dy = αy0.

Then xOd+ yOd⊇ _α1(x0Od+ y0Od). Since the inverse of

a b c d

is still in Γd, also _β1(xOd + yOd) ⊆ x0Od + y0Od holds. Then

[xOd+ yOd] = [x0Od+ y0Od] in the class group.

If [xOd + yOd] = [x0Od+ y0Od], we want to prove that exists a

γ as in the the thesis. We will start proving that such γ exists between principal ideals; we will do this proving that any principal [xOd+ yOd]can be moved to [1, 0]. Since [xOd+ yOd]is principal,

there exists a constant in Kd such that αxOd+ αyOd= Od. If this

holds, notice that αx and αy are necessarily integers: if they are in Kdr Od, then αxOd+ αyOd cannot be Od. Furthermore, there

exist i, j ∈ Od such that

αix − αjy = 1 − αx;

then the matrix

αx j αy 1 + i

is in P SL(2, Od) and moves [1, 0] to [αx, αy] = [x, y].

For the general case, we just need to conjugate for a matrix in P SL(2, Kd)to bring one pair to [1, 0]: the other element (being in

the same equivalence class in the class group) will be moved to a principal ideal; then we can use the equivalence between principal ideals.

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Corollary 1.47. Since the cusp set is a commensurability invariant, if Γ is commensurable with Γd, then the number of ends of H

3 /Γ is P 1_(K d)/ Γ . We state here a couple of results that we will need in the next chapters. Their proofs are either very similar to some arguments already employed in the previous pages, or go too far beyond the aim of the thesis. Therefore we omit them.

Fact 1.48. Let I ⊂ Od be an ideal, with I = P1a1P a2 2 . . . Pnan. Then |P SL(2, Od_{/I)| =}        6 if N(I) = 2 N (I)3Q P |I 1 − _{N (P )}1 2 if I = h2i N (I)3 2 Q P |I 1 − _{N (P )}1 2 otherwise

Theorem 1.49. Suppose that hd= 1, and let a, b, x, y ∈ Od satisfy ha, bi =

hx, yi = Od. Then a_b is equivalent to x_y (as elements of ∂H3) modulo the

action of Γ(I) < P SL(2, Od) if and only if

a b = kx y mod I, where k is a unit in Od.

1.5 References

The material discussed in section 1.1 and subsection 1.1.1 can be found in many classical books: [Mar16] in Chapter 1-2 is straight and self-contained, also in [MR02] there is an introduction in Chapter 1. More insight and a proof of the topics in subsection 1.1.2 can be found in [Mar16] in Chapter 4 or in [BP92] in Chapter D; a more algebraic point of view can be found in [Joh06] in Chapter 12. Subsection 1.1.3 can be found in [Mar16] in Chapter 4 or in [MR02] in Chapter 5 and 12; for further information about systoles (that we will need in the next of the thesis) [AR00] can be consulted. Theorem in 1.1.4 is stated and proved in [Mar16] in Chapter 15.

The arguments in section 1.2 are standard topic in algebraic number theory and can be found in any classical book about this argument, such as [Mar77] or [KI90]; [MR02] contains in Chapter 0 a nice but fast recap about these notions.

Sections 1.3 and 1.4 are a selection of theorems and denition from [MR02]; the last facts can be found in [BR14] and [BR17].

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Chapter 2 Principal Congruence Link

Groups

In this chapter we introduce the principal congruence manifolds and we give the ideas of the constructions and the methods that were employed in order to obtain the complete classication of the principal congruence manifolds that are link complements. Let us start with the denitions.

Given an ideal I of Od, the principal congruence group associated with I

is dened as:

Γ(I) = Ker{P SL(2, Od) → P SL(2, Od/I)}.

A manifold M = H3_{/Γ is a principal congruence manifold if Γ is} conju-gated to Γ(I) for some I ideal of Od. Our aim is to identify the ideals I for

which H3_{/Γ(I) is homeomorphic to a link complement in S}3. What we want

to do rst is to prove that this is possible only for nitely many ideals I, and then to nd out which of them are link complements.

2.1 Exclusion criteria

In this section we will nd properties that have to be satised by d and I in order to let H3_{/Γ(I) be a link complement.}

2.1.1 Cuspidal cohomology

The result we want to present is about the values of d which give rise to ideals I whose associated manifold is a link complement. The main theorem of this discussion is

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Theorem 2.1. Let I be an ideal of Od and H 3

/Γ(I) be a link complement. Then

d ∈ {1, 2, 3, 5, 6, 7, 11, 15, 19, 23, 31, 39, 47, 71}. We only sketch an idea of the proof.

Let H3_{/Γ(I) be a non-compact nite-volume hyperbolic manifold. It} fol-lows from Theorem 1.10 that it is homeomorphic to the interior part Int(M) of a compact manifold M with boundary ∂M consisting of tori. We can then consider the long exact sequence with rational coecients

... → Hn(M, ∂M )−→ Hαn n_{(M )}₋_{→ H}in n_{(∂M ) −}_{→ H}n+1_{(M, ∂M ) → ...}

The image of α∗ _{in this sequence is called the cuspidal cohomology of}

Γ(I). What we want to prove is the following:

1. the function α2 can be trivial only for a nite number of values of d;

2. if H3_{/Γ(I) is homeomorphic to S}3

r L, then α2 is trivial.

For the part 1 we will need some results about Zimmert sets and their properties. Let D be the discriminant of Q(√d).

Denition 2.2. The Zimmert set Zd is the set of all the positive integers n

such that the following conditions are satised: 4n2 _{< |D| − 3} and n 6= 2;

D is a quadratic non-residue modulo all odd prime divisors of n; n is odd if D 6≡ 5 mod 8

Let z(d) be the cardinality of the set Zd.

Fact 2.3. For all but a nite number of values of d, we have z(d) ≥ 2. The signicance of this set is that we can split a specic fundamental domain F for Γd into z(d) portions so that for each j ∈ Zd there exists

a function ϕj : F → S1 which equals 1 outside the j-th portion and that

induces a surjection ϕj∗ : Γd → π1(S

1_{) = Z. These functions can be combined}

in order to get a map f : H3_/Γ_d → W

1≤i≤z(d)S 1

i, where W1≤i≤z(d)S 1

i denotes

the wedge of z(d) copies of S1, such that

f∗ : Γd→ Zz(d)

is surjective. By studying the behavior of f∗ on parabolic elements, one can

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the translation element τw : z → z + w in the stabilizer of ∞, where w

generates Od together with 1 as in Fact 1.43, has a non-trivial image

λ, that can be chosen to be a primitive element in Zz(d);

the other generator in the stabilizer of ∞ is sent to a power of λ (because its image commutes with λ);

parabolic elements that x all the other equivalence classes of cusps have trivial image.

So we can obtain a function: ˜ f : Γd f − → Zz(d) π−→ Zz(d) hλi∼= Zz(d)−1 that is onto and which is zero on all parabolic elements.

Recall Theorem 1.46. For each i ∈ P1(Kd)/

Γd let Pi(Γd) denote the

max-imal parabolic stabilizer of a representative cusp: it consists in the Z ⊕ Z subgroup that xes it. Let

P (Γd) =

Y

i∈P1_(K d)/Γd

Pi(Γd).

There is a natural function

α(Γd) : P (Γd) i

− → Γab_d , which can be tensorized by Q to obtain

α(Γd)Q : P (Γd) ⊗ZQ → Γ ab

d ⊗ZQ.

Observation 2.4. We have a natural map ˜f∗ : Γab

d ⊗Z Q → Q

⊕z(d)−1_{. If}

z(d) ≥ 2, we can consider the composition: P (Γd) ⊗ZQ

α(Γd)Q

−−−−→ Γab_d ⊗_Z_Q f˜

∗

−→ Q⊕z(d)−1

The composition of these two functions is trivial, but the second one is sur-jective. Then α(Γd)Q cannot be surjective.

Now let Γ be a torsion-free subgroup of nite index in Γd (for example,

let Γ = Γ(I) the fundamental group of the manifold we want to study). In the same way as above, we have a map

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and since Γ is of nite index in Γd, the mapping

π : Γab⊗_Z_{Q → Γ}ab d ⊗ZQ

is surjective. It is easy to see that the following diagram commutes: P (Γ) ⊗_Z_Q P (Γd) ⊗ZQ Γab⊗_Z_Q Γab_d ⊗_Z_Q α(Γ)_Q ψ π α(Γd)Q

where ψ is dened up to a choice of a cusp representative in P1(Kd)/

Γd for

each element in P1(Kd)/

Γ. If α(Γd)Q is non-surjective (for example, as in

Observation 2.4), so neither is α(Γ)Q.

Let's see how this help us with the cohomology sequence. The key concept here is that, as a consequence of Theorem 1.10, the function α(Γd)Qis exactly

the natural map (induced by inclusion) H1(∂M ) → H1(M ). If the latter is

not surjective then, by duality, neither is the map H1_{(∂M ) → H}2_{(M, ∂M )}_.

This implies that α2 _{is not trivial, thus concluding the part 1.}

For the proof of the part 2, let us suppose that H3_{/Γ(I) = S}3

r L, and let m be the number of components of L. It is easy to prove, via the Lefschetz duality and the Mayer-Vietoris sequence, that

H2_{(M ) = Q}⊕m−1; H2_{(∂M ) = Q}⊕m; H3(M, ∂M ) ∼= H0(M ) = Q.

Then the map ψ is the following sequence needs to be injective: H2(M, ∂M )−→ Hα2 2_{(M )}₋_{→ H}ψ 2_{(∂M )}₋_{→ H}ϕ 3_{(M, ∂M ).}

As a consequence, α2 _{is trivial.}

This concludes our sketch of the proof of Theorem 2.1. We notice that in the complete proof of this theorem further constructions than Zimmert sets were used in order to compute the non-trivialness of the cuspidal cohomology for Γd.

2.1.2 Bound on norm of the ideal I

Once we limited the number of the values of d where we should search for, we need to do the same for the ideals I ⊂ Od. The theorem which gives us

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Theorem 2.5. Let H3_{/Γ(I) be homeomorphic to a link complement in S}3.

Then N(I) < 39.

Corollary 2.6. There are only nitely many ideals I for which H3_{/Γ(I) can} be a link complement in S3_.

Proof. The result follows from Lemma 1.26.

Recall the systole length sl(M) from Subsection 1.1.3. The rst step in order to prove the Theorem 2.5 is this lemma:

Lemma 2.7. Let L be a link in S3 _{whose complement admits a complete}

hyperbolic structure of nite volume. Then sl(S3

r L) < 7.17. Proof. See [AR00] and [BR17].

Then we need to prove this:

Lemma 2.8. Let γ ∈ Γ(I) be a hyperbolic element. Then tr(γ) ≡ ±2 mod I2.

Proof. The element γ is equal to

±1 + a b c 1 + d

with a, b, c, d, ∈ I (this is because γ ∈ Γ(I)), where det γ = 1. Then 1 + a + d + ad − bc = 1 ⇒ a + d = bc − ad.

Therefore tr(γ) = ±(2 + a + d) = ±(2 + bc − ad) ≡ ±2 mod I2.

Now we just need to do a computation: let γ ∈ Γ(I) be the hyperbolic element corresponding to the minimal length geodesic. Recall, from Lemma 1.12, that cosh(`(γ)/2) ≥ |tr(γ)|/2. Then

( tr(γ) = ±2 + α, α ∈ I2 `(γ) < 7.17 ⇒ (tr(γ) = ±2 + α, α ∈ I2 2 arccosh|tr(γ)|₂ < 7.17 ⇒ ⇒ ( tr(γ) = ±2 + α, α ∈ I2 |tr(γ)| < 2 cosh 7.17 2 ' 36.2 .

Hence we deduce that there exists a β in I such that |β| < √39, and therefore N (I) < 39.

Observation 2.9. There exist non principal ideals I for which N(I) < 39 but the squared norm of every x in I is greater than 38. We should keep in mind that we proved that there exists an element in I such that its squared norm is lower than 39.

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2.2 Inclusion criteria

We now want to discuss the main method that can be used to prove that H3_{/Γ(I) is homeomorphic to a link complement. We rst recall some general} facts and briey discuss them; let L = L1∪ . . . ∪ Lm be a link in S3 and X(L)

be the complement of an open tubular neighborhood of L. Then: 1. If Int(X(L)) ∼= S3 r L is homeomorphic to H

3

/Γ(I) (so that Γ(I) ∼= π1(S3 r L)), then Γ(I)ab ∼= Z⊕m and Γ(I) is generated by parabolic elements. The rst fact is an easy consequence of the Wirtinger pre-sentation: every relation of the type xjxhx−1j = xk becomes, under the

eect of the abelianization, xh = xk. Then we can simplify the set of

generators keeping one element per link component and deleting these relations. At the end we obtain something like

π1(S3r L)ab = hx1, . . . , xm|[xi, xj] = 1i = Z⊕m.

The second stated fact holds because the elements in the Wirtinger pre-sentation correspond to parabolic elements, as a consequence of Theo-rem 1.10.

2. The manifold H3_{/Γ(I) is homeomorphic to S}3

r L if and only if Γ(I) can be trivialized setting one parabolic element for each cusp of Γ(I) equal to 1. The "only if" part is trivial: operating the trivial Dehn lling on each component of L (i.e. setting as 1 one element for each cusp, that is a parabolic element) the fundamental group becomes triv-ial because it becomes π1(S3). Conversely, given Perelman's resolution

of the Geometrization Conjecture, if Γ(I) can be trivialized by set-ting one parabolic from each cusp of Γ(I) equal to 1, then H3_{/Γ(I) is} homeomorphic to a link complement in S3 _.

2.2.1 Operative method

The operative method to nd out if H3_{/Γ(I) is a link complement is then} divided into two steps:

Check if Γ(I) is generated by parabolics.

Let Γ(I) < P SL(2, Od), and let Pibe the subgroup of P SL(2, Od)xing

the cusp ci for i = 1, . . . , hd. Let Pi(I) = Pi ∩ Γ(I) be the subgroup

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of {P1(I), . . . , Phd(I)}. Note that H

3

/Γ(I) can have more than hd ends,

but their subgroups are still conjugated under the action of P SL(2, Od)

so we can consider the normal closure of {P1(I), . . . , Phd(I)} without

losing parabolic elements. Clearly Nd(I) < Γ(I), because Γ(I) is

nor-mal in P SL(2, Od). Then Γ(I) is generated by parabolics if and only if

Nd(I) = Γ(I). This is usually done by Magma computation.

Find parabolic elements in Γ(I), one for each cusp, so that by killing these elements we get the trivial group.

After nding a full set of subgroups that x cusps (adding the con-jugated we talked about in the previous step), it remains to nd one parabolic element from each of these groups with the property above.

2.3 Some examples

In the last section we have spotted one clear way to proceed in order to prove that a given manifold H3_{/Γ(I) is homeomorphic to a link complement in S}3_.

We now provide some examples of how this can be done.

2.3.1 d = 1; I = h2 + ii.

What we want to prove now is that H3_{/Γ(h2 + ii) is a link complement. We} divide the proof in several steps.

Γ(h2 + ii) is generated by parabolic elements.

Clearly N(h2 + ii) = 5, then, by Fact 1.48, the index of Γ(h2 + ii) in P SL(2, O1) is 60.

We want to nd the order of P∞, that is the stabilizer of ∞ in P SL(2, O1).

Notice that h1 = 1, so there is only one cusp in H 3

/

P SL(2, O1).

The elements in P SL(2, O1) which stabilize ∞ are of the form

a b 0 a−1

,

then, up to multiplication by −1, the element a needs to be in {1, i} and b can assume all the possible values in h2 + ii, which means that we have 2 · 5 = 10 elements in the image of the stabilizer of ∞. Then the number of cusps of H3_{/Γ(h2 + ii) is 60/10 = 6.}

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We x the following notation: t =1 1 0 1 ; u =1 i 0 1 .

Now we prove that P∞(h2 + ii), that is dened as P∞∩ Γ(h2 + ii), is

equal to ht2_{u, t}5_i_{. We know that every element in P}

∞(h2 + ii) is of the form a b 0 a−1 ,

where a ≡ 1 mod h2+ii and b ≡ 0 mod h2+ii, from which we obtain that a = 1 and b = (α+iβ)(2+i), where α, β ∈ Z. Given such a matrix, it is easy to prove that it can be obtained as a product (t5₎−β_·(t2_u)α+2β_.

The result follows.

At this point a Magma computation shows that the index of the normal closure of P∞(h2 + ii)in P SL(2, O1)is 60:

[P SL(2, O1) : hP∞(h2 + ii)i] = 60.

It follows that such normal closure is equal to Γ(h2 + ii).

G<a,l,t,u>:=Group<a,l,t,u | l^2, a^2, (t*l)^2, (u*l)^2, (a*l)^2, (t*a)^3, (u*a*l)^3, (t,u)>;

h:=sub<G | t^2*u, t^5>; n:=NormalClosure(G,h); print Index(G,n); \\60

This presentation of G = P SL(2, O1) is a consequence of the work

in [Swa71] and [Pag12]. Before studying the next step, we make the following observation:

Observation 2.10. Let hd be the class number of Kd. We want to

know the number of the orbits of the action of Γ(I) on P1_(K

d)

(remem-ber Corollary 1.47). We know that, with respect to the action of Γd,

this number is hd. The subgroup Γ(I) is contained in Γ and has index

PSL(2, O d_/I)

= q. Through the mapping

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a cusp stabilizer Γζ is mapped to a subgroup of order l. Then

[Γζ : StabΓ(I)(ζ)] = l.

It follows that the orbit of ζ splits in q

l orbits. If for every equivalence

class of cusps ζ the stabilizer Γζ is mapped to a subgroup of order l,

then the number of cusps of H3_{/Γ(I) will be h}dq_l.

Now we want to nd one element per cusp such that by killing those elements we obtain the trivial group (or, equivalently, the group is normally generated by those elements.). This is done in two parts: the rst one is achieved by nding six elements in P1_(K

1) which are

inequivalent under the action of Γ(h2 + ii); the second one by nding one element for each of these cusps that have the property we already stated.

The set of elements is S = {0, ∞, ±1, ±2}. The fact that they are mutually inequivalent is an application of Theorem 1.49. The parabolic elements that x them are, respectively,

t2u, at2ua, t−1at2uat, tat2uat−1, t−2at2uat2, t2at−3uat−2. The fact that they generate hP∞(h2 + ii)i can be checked adding the

following rows at the Magma routine we showed above: r:=sub<n|t^2*u, a*t^2*u*a, t^-1*a*t^2*u*a*t,

t*a*t^2*u*a*t^-1,t^-2*a*t^2*u*a*t^2,t^2*a*t^-3*u*a*t^-2>; print Index(n,r);

\\1

2.3.2 d = 15; I = h3, 1 + w

−15

i

.

In this case N(I) = 3, and it can be easily checked that P SL(2, Z/3_Z) ∼= P SL(2,Od/I) is isomorphic to the alternating group A4. We x the following

presentation for P SL(2, O15):

ha, t, u, c|a2 _{= (ta)}3 _{= ucuatu}−1

c−1u−1a−1t−1 = 1, [t, u] = [a, c] = 1i. Even if we will not use this, we also give the explicit matrix form of these elements, in order to understand what is going on:

a =0 −1 1 0 ; t =1 1 0 1 ; u = 1 w−15 0 1 ; c = 4 1 − 2w−15 2w−15− 1 4 .

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In H3_/Γ₁₅ we have two orbits of cusps, this being a list of a representative for each one with the corresponding stabilizer:

(∞, P1 = ht, ui), 1 +√−15 4 , P2 = huca, c −1 au−1c−1u−1tai .

Each of these subgroups maps to a subgroup of order 3 in P SL(2, Od_/I)

and then, by Observation 2.10, every cusp orbit for Γ15 splits in 4 cusp orbits

for Γ(I). Hence, there are 8 cusp orbits.

Let us call Pi(I) = Pi∩ Γ(I). It can be checked that P1(I) = ht3, tui and

P2(I) = h(uca)3, (c−1au−1c−1u−1ta)i, by using the identity

[Pi : Pi(I)] = 3.

Let h be the element ata(at)−1_{. Magma can check that}

[hΓ(I), a, hi : Γ(I)] = 4,

then we can consider the conjugate of P1(I) and P2(I) by {Id, a, h, ah} and

we obtain a full set of stabilizers, one for every cusp. We then take one element in each of these subgroups:

{t−2u, atua−1, ht−2uh−2, (ha)tu(ha)−1, (c−1au−1c−1u−1ta)(uca)3,

a(c−1au−1c−1u−1ta)a−1, h(c−1au−1c−1u−1ta)h−1, ha(c−1au−1c−1u−1ta)(ha)−1}, and check with Magma that, trivializing these elements, the group is trivial-ized. Thus Γ(I) is a 8-component link complement. The following rows are the ones necessary to check the statements we talked about:

G<a,c,t,u>:=Group<a,c,t,u|(t,u),(a,c),a^2, (t*a)^3,u*c*u*a*t*u^-1*c^-1*u^-1*a*t^-1>; H:=sub<G|t^3,t*u,(c^-1*a*u^-1*c^-1*u^-1*t*a),(u*c*a)^3>; N:=NormalClosure(G,H); print Index(G,N); 12 \\

Here we dened G = Γ15; we can check that H is contained into Γ(I), hence

N is contained in Γ(I); by the fact that they have the same index, we can conclude that N = Γ(I).

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h:=a*t*a*t^-1*a; A:=sub<G|N,a,h>; print Index(A,N); 4

\\

This is exactly what we said before, with the same notation.

Q:=quo<N| t^-2*u, a*t*u*a, h*t^-2*u*h^-1, h*a*t*u*a*h^-1, (c^-1*a*u^-1*c^-1*u^-1*t*a)*(u*c*a)^3, a*(c^-1*a*u^-1*c^-1*u^-1*t*a)*a, h*(c^-1*a*u^-1*c^-1*u^-1*t*a)*h^-1, h*a*(c^-1*a*u^-1*c^-1*u^-1*t*a)*a*h^-1>; print Order(Q); 1 \\

And this is the last check we needed.

2.4 References

The material discussed in section 2.1.1 is presented in [MR02] in Chapter 9, where more insight about the connection of link complements with arithmetic manifolds can be found.

The arguments in section 2.1.2 are pointed out in [BR14], with results from [AR00].

The method and examples in section 2.2 can be found in [BR14] and [BR17], where a lot of other ideas to point out particular cases are introduced.

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Chapter 3 Regular tessellations

In this chapter we want to present the work exposed by Goerner in [Goe15]. There the author used regular tessellations to classify which principal con-gruence manifolds are link complements for d = 1 and d = 3.

Denition 3.1. A regular tessellation is a hyperbolic 3-manifold tessellated by ideal Platonic solids such that the group of isometries acts transitively on oriented ags, where a ag is a sequence of j-dimensional faces fj

f0 ⊂ . . . ⊂ f3.

An elegant way to encode the information of such a tessellation is the Schlä notation: by {p, q} we mean a polyhedron where the faces are p-gons and q of them meet at each vertex; the symbol {p, q, r} denotes the tessellation of the hyperbolic space in polyhedra of type {p, q}, where r of them meet at each edge. If we consider tessellations of the euclidean space, the tessellation of type {4, 3, 4} is the most natural one: made by cubes.

Only for certain values of {p, q, r} we obtain regular ideal tessellations of the hyperbolic space:

{3, 3, 6} tessellated by tetrahedra, {3, 4, 4} tessellated by octahedra, {4, 3, 6} tessellated by cubes,

{5, 3, 6} tessellated by dodecahedra;

see [Mar16], Chapter 3 for details. For normalization purposes, we move each of the above regular ideal tessellations in H3 _{such that there is a face}

with three consecutive vertices at the points ∞, 0, and 1. For the values 33

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of {p, q, r} described above, we denote by Γ{p,q,r}

⊂ P SL(2, C) the set of orientation preserving isometries of the regular tessellation. We now give another denition that can be proved to be equivalent to the previous one. Denition 3.2. A manifold M is a regular tessellation of type {p, q, r} if it is a quotient M = H3_{/H, where H is a torsion-free normal subgroup of} Γ{p,q,r}.

Let now z be a number of the form a + bu where u = e2πi

r and a, b ∈ Z.

When r = 6 or r = 4 the element pz =

1 z 0 1

is in Γ{p,q,r} _{for every z. Denote by U}{p,q,r}

z the normal closure of pz in Γ{p,q,r}.

Denition 3.3. The universal regular tessellation of cusp modulus z is the quotient

U_z{p,q,r} = H

3

Uz{p,q,r}

.

Notice that if z0 _{= zu}_{, then U}{p,q,r} z = U {p,q,r} z0 , because p_z = gp_z0g−1where g =u 0 0 1 ∈ Γ{p,q,r}. Furthermore, if z0 _{= z} _{then manifold U}{p,q,r}

z changes only its orientation.

Therefore, studying universal regular tessellations, we can consider only the elements z in the following canonical form:

Denition 3.4. z = a + bu is in canonical form if a ≥ b ≥ 0.

The connection between these objects and principal congruence manifolds is the following:

Fact 3.5. These equalities hold: P GL(2, O1) = Γ{3,4,4}.

P GL(2, O3) = Γ{3,3,6}.

Moreover, P SL(2, Oi) is an index 2 subgroup of P GL(2, Oi) for i = 1, 3.

It is hence evident a connection between principal congruence manifolds and regular tessellations: if we classify regular tessellations which are link complements, we will obtain results about our main purpose.

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Figure 3.1: Fundamental domain for the regular tessellation T2+ζ in the case

i = 3. When i = 4, we have squares instead of triangles. Image source: [Goe15].

3.1 Cusp modulus

In this chapter we clarify the notion of cusp modulus introduced in Denition 3.3.

3.1.1 Tessellations of the torus

For i = 1, 3, consider Oias a subset of C. We can tessellate the complex plane

by drawing a segment connecting each pair of elements in Oi at distance one.

Let now hzi ⊂ Oi be a non-zero ideal. It is possible to show that C/hzi is a

torus and so we obtain a tessellation of the torus.

Denition 3.6. Given z ∈ Oi, let Tz be the triangulation of the torus

obtained as a quotient of the above regular tessellation of C by the action of the elements in the ideal hzi by translations. We denote by T∗

z its dual

tessellation, dened by taking the center of each polygon as a vertex and joining the centers of adjacent polygons.

Figure 3.1 contains an example of such a triangulation. It is known that every regular tessellation of the torus is of the form Tz or Tz∗; furthermore,

we notice that if u is a unit in Oi then Tz = Tzu, and that Tz = Tz if and

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3.1.2 Cusp modulus

We now want to associate a cusp modulus hzi to any regular tesselation. In order to do this, we rst do a basic observation about ideal regular tessella-tions of type {p, q, r} in H3: recall our normalization and take a horosphere

O near ∞; this plane is tessellated by its intersections with the polyhedra. By denition, there are q edges that meet at each vertex, and hence O is tessellated by regular q-gons.

A {p, q, r} regular tessellation M of a nite-volume oriented cusped hy-perbolic manifold induces, for each cusp, an oriented tessellation {q, r} of the boundary of the cusp neighborhood, that is a torus. Since the tessellation of the 3-manifold is regular, the induced tessellation on the tori is regular and it is the same for each cusp. Therefore we can associate to a regular tessellation an ideal hzi in O1 or O3 (depending on q); we will call it the cusp

modulus of M.

From an algebraic viewpoint, if M is a regular tessellation of the form H3_{/H, where H is a torsion-free normal subgroup H of Γ}{p,q,r}_{, the cusp}

modulus depends only on the elements pw =

1 w 0 1

contained in H. In particular, the set of elements above the diagonal of these matrices is an ideal of O1 or O3, and then (these being PIDs) it is generated

by an element z, which can be proved to be the cusp modulus. Two easy but very important consequences are the following:

Lemma 3.7. Let M = H3_{/H be a (not necessarily nite volume) regular} tessellation {p, q, r} with cusp modulus z. Then there is a covering map Uz{p,q,r} → M.

Proof. The group H contains pz and is normal in Γ{p,q,r}. Thus it must

contain U{p,q,r}

z by denition.

Lemma 3.8. If a regular tessellation M is a link complement, then it is also a universal regular tessellation U{p,q,r}

z .

Proof. A regular tessellation M is universal if and only if it is generated by parabolic elements. If M is a link complement, by the Wirtinger presentation it also has this property.

Principal congruence link complements

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale