数理解析研究所講究録
第 1571 巻 2007 年 31-58

31

The geometry and topology of arithmetic hyperbolic
3-manifolds
Alan W. Reid*

May 12, 2007

1

Introduction

This paper is based on three lectures given by the author at the RIMS Symposium, “Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces” held at the Research Institute for Mathematical
Sciences, Kyoto University, in December 2006. The goal of the $lecture8$ was to describe recent workon understanding topological and $g\infty metric$ properties of arithmetic hyperbolic 3-manihl&, andthe connections with number theory. This is the theme of the paper.
Our discussion of topological aspects of arithmetic hyperbolic 3-orbifolds is motivated by thefollowing central conjectures from 3-manifold topology:

Conjecture 1.1. Let $M$ be a closed hyperbolic S-manifold, then $M$ is mrtually Haken; $|eM$ has afinite sheeted cover $N$ which contains embedded incompressible surface (necessarily of genus at least2).

Conjecture 1.2. Let $M$ be a closed hyperbolic S-manifold, then $M$ has a finite sheeted cover $N$ forwhich $b_{1}(N)>0$ (where $b_{1}(N)$ is the first Betti number of $N$).

We can also rephraee Conjecture 1.2 to say that $vb_{1}(M)>0$ where $vb_{1}(M)$ is defined by$vb_{1}(M)=\sup${$b_{1}(N)$:$N$

is a finite cover of $M$ },and is called the virtual first Betti number of $M$ .

Conjecture 1.3. Let $M$ be a closed hyperbolic S-manifold, then $vb_{1}(M)=\infty$ .

Conjecture 1.4. Let $M$ be a dosed hyperbolic S-manifold, thenis large; that is to say, some
admits ahomomorphism onto a non-abelian fiee group.
finite dex subgroup of$\pi_{1}(M)$$|n$$\pi_{1}(M)$$su\dot{\eta}e\iota\hslash ve$

Now it is clear that Conjecture 1.4 implies Conjecture 1.3 implies Conjecture 1.2, and standard
3-manifold topology shows that Conjecture 1.2 implies Conjecture 1.1. Our interest here is in recent
work towards reversing these implications.
Our $g\infty metric$ discussion is centered around the set of lengths of closed geodesics, as well as theset of geodesics themselves, and in particular on how these force a certain rigidity on commensurability classes. For example, the length spectrumof a hyperbolic 3-manifold $M$ is the set of alllengths of closed geodesics on $M$ counted with multiplicities. A question that has attracted someattention is whether hyperbolic 3-manifolds with the same length spectra are commensurable. We
discuss this and other related questions in this paper.
$\mathcal{L}(M)$$\overline{*\supportd}$
in part by the NSF

32Given these preliminary remarks, the paper is organized as follows. In \S 2 we recall the definitionof arithmetic Fuchsian and Kleinian groups, arithmetic hyperbolic 2 and 3-orbifolds together withsome other terminology and notation that we will use. In \S 3, we discuss a geometric characterizationof arithmetic hyperbolic 3-manifolds which will be used in \S 4 to show that for arithmetic hyperbolic
3-manifolds, Conjecture 1.2 implies Conjecture 1.4. In \S 4, we also discuss how certain conjecturesfrom automorphic forms implies Conjectures 1.1-1.4. This is well-known to the experts, and our
purpose is simply to sketch some of the ideas involved. In the opposite direction, in \S 6 we discuss how
widely distributed arithmetic rational homology 3-spheres are. We motivate this in \S 5 by discussing
a similar question in dimension 2 that offers useful comparisons for dimension 3. Finally, in \S 7 wediscuss the $g\infty metric$ properties of arithmetic hyperbolic 2 and 3-orbifolds.
Acknowledgements: The author would like to thank Michihiko Fuji for organizing the symposium
and his hospitality during the symposium. He also wishes to thank his various co-authors of theresults discussed here, particularly D. D. Long. In addition, the author would like to thank F.
Voloch for a helpful conversation, and he is very grateful to F. Calegari for helpful discussion andcorrespondence regarding the proof of Thrrem 4.14.

2

Preliminaries

For more details on the topics covered in this section see [29] for number $th\infty retic$ background and[39] for detaik on quaternion algebras and arithmetic groups.

21
By a number field we mean a finite extension of Q. The ring of integers of will be denoted. A place of. will be one of the canonical absolute values of . The finite places of. An infinite place of is either real, corresponding
correspond bijectively to the prime ideals of
to an embedding of into $R$ , or complex, corresponding to a pair of distinct complex conjugate
embeddings of into C. We denote by
the completion of of at a place . When is an infinite
place,
is isomorphic to $R$ or depending on whether is real or complex.
, the norm of $A$ is the cardinality of the quotient ring $R_{k}/A$ and will be
If $A$ is an ideal ofdenoted $NA$ .
$k$$R_{k}$

$\nu$

$k$

$k$

$k$

$k$

$k$

$R_{k}$

$k$

$k$

$k$

$k_{\nu}$$C$$k_{\nu}$

$\nu$

$\nu$

$\nu$

$R_{k}$

2.2
be a field of characteristic different from 2. The standard notation for a quaternion algebra
over is the following. Let and be non-zero elements of , then(known as the Hilbert
Symbol) denotes the quaternion algebra over with basis $\{1, i,j, ij\}$ subject to $i^{2}=a,$ $j^{2}=b$ and$ij=-ji$ .
Let be a number field, and a place of . If is a quaternion algebra defined over , theclassification of quaternion algebras
over the local fields
is quite simple. If iscomplex thenis isomorphic to $M(2, k_{\nu})$ over . Otherwise there is, up to isomorphism over ,a unique quaternion division algebra over , andis isomorphic overto either thls division
algebra or to $M(2, k_{\nu})$ .
Let be a quaternion algebra over the number field .
is ramified at a place of if
is a division algebra. $Otherwi_{8}e$ we say
is unramified at . We shall denote the set of places(resp. flnite places) at which $B$ is ramifled by Ram (resp. $Ram_{f}B$ ). The discriminant of $B$ is the-idealis the prime ideal associated to the place .
where
Let

$k$

$k$

$k$

$b$

$a$

$(\not\simeq)$

$k$

$k$

$B$

$k$

$\nu$

$k$

$B_{\nu}=B\otimes_{k}k_{\nu}$$B_{\nu}$$k_{\nu}$$\nu$$k_{\nu}$$k_{\nu}$$k_{\nu}$$B_{\nu}$$B$$k_{\nu}$

$k$

$B$

$B$

$\nu$

$\nu$

$B$

$R_{k}$$\prod_{\nu\in Ram_{t}B}\mathcal{P}_{\nu}$$\mathcal{P}_{\nu}$

$\nu$

$k$

$B_{\nu}$

33

We summarize for convenience the classification theorem for quaternion algebras over number
fields (see [39] Chapter 7).

.

Theorem 2.1.

.

$\bullet$The set Ram$B$is finite, of even cardinality and contains no complex places.

Conversely, suppose
is a finite set of places of which has even cardinality and whichcontains no complex places. Then there is a quatemion algebra $B$ over with Ram $B=S$ ,and this $B$ is unique up to isomorphism over .
$S$

$k$

$k$

$k$

$B$

is a dimsion algebra of quatemions if and only if Ram$B\neq\emptyset$

.

$O$

2.3
We next recall the definition of arithmetic Fuchsian and Kleinian groups.
Let be a totaUy real number field, and let be a quaternion algebra defined over which isramified at all infinite places except one. Let $\rho:Barrow M(2,R)$ be an embedding, be an order of, andthe elements of norm one in . Then $P\rho(\mathcal{O}^{1})<PSL(2, R)$ is a finite co-area Fuchsiangroup, which is co-compact if and only if $B$ is not isomorphic to $M(2, Q)$ . A Fuchsian group isdefined to be arithmetic if and only if is commensurable with some such
.
$k$

$B$

$k$

$O$

$B$

$O^{1}$$O$$\Gamma$$P\rho(O^{1})$$\Gamma$

Notation: Let $B/k$ be as above and$O$

be an oder of B. We Utll denote the group$P\rho(\mathcal{O}^{1})$by$\Gamma_{\mathcal{O}}^{1}$.

Arithmetic Kleinian groups are obtained in a similar way. In this case we let be a number field
having exactly one complex place, and a quaternion algebra over which is ramified at all realplaces of . As above, if is an order of and $\rho:O^{1}rightarrow SL(2, C)$ , thenis a Kleinian group offinite co-volume. An arithmetic Kleinian group is a subgroup of $PSL(2, C)$ commensurable witha group of the type . An arithmetic Kleinian group is cocompact if and only if the quaternion
algebra as above is a division algebra.
In both the Fuchsian and Kleinian cases, the isomorphism class of the quaternion algebra $B/k$
determines a wide commensurability class of groups in $PSL(2, R)$ and $PSL(2, C)$ respectively (see[39] Chapter 8). By Theorem 2.1 the isomorphism classes of such quaternion algebras $wiU$ becompletely determined by the finite set of places of at which is ramified.
A hyperbolic orbifold
orwill be called arithmetic if is an arithmetic Fuchsian or
Kleinian group.
Recall that if is a Kleinian group, the invariant trace-field of is the field
$k$

$B$

$O$

$k$

$k$

$B$

$\Gamma_{O}^{1}$$\Gamma$$\Gamma_{\mathcal{O}}^{1}$

$B$

$k$

$H^{2}/\Gamma$$B$$H^{3}/\Gamma$$\Gamma$$\Gamma$$\Gamma$$k\Gamma=Q(\{tr^{2}\gamma:\gamma\in\Gamma\})$and the invariant quaternion algebra$A\Gamma=\{\Sigma a_{j}\gamma_{j} :a_{j}\in k\Gamma,\gamma_{j}\in\Gamma^{\{2)}\}$.

As discussed in [39] these are invariants of commensurability. When is arithmetic, the field
and algebra $B$ coincide withandArithmetic Fuchsian or Kleinian groups form a small but interesting subclass of Fuchsian or
Kleinian groups. For example it is known that there are only finitely many conjugacy classes ofarithmetic Kleinian groups whose volume is bounded above by some constant $K$ . A similar statement
holds for arithmetic FMchsian groups and this also shows that there are only finitely many conjugacy
classes of arithmetic Nchsian groups of a fixed signature (see [39] Chapter 11).
$k$$\Gamma$$k\Gamma$$A\Gamma$

Notation: We shall call a group is derived $Pom$ a quatemion dgebra if, regardless of whether has finite index in .
$G$$\Gamma_{\mathcal{O}}^{1}$$G$$\Gamma_{\mathcal{O}}^{1}$$G$

is a subgroup of some

34

2.4
We set up some notation and collect some information on 3-orbifolds that we will make use of.
Let be a compact orientable 3-orbifold. We shall denote by sing $(O)$ its singular locus, and $|Q|$
the underlying 3-manifold. Let $sing^{0}(Q)$ and $sing^{-}(Q)$ denote the components of the singular locus
with, respectively, zero and negative Euler characteristic. For any prime , let $sing_{p}(Q)$ denote theunion of the arcs and circles in sing $(Q)$ with singularity order that is a multiple of . Let
and $sing_{p}^{-}(Q)$ denote those components of $sing_{p}(Q)$ with zero and negative Euler characteristic.
When $Q=H^{3}/\Gamma$ is a closed orientable hyperbolic 3-orbifold the nature of the singular locus
is completely understood. is a link or graph in $|Q|$ and the local groups are either cycllc, dihedral,or . Note the dihedral group of order 4 is $Z/2ZxZ/2Z$ . The cases of the dihedral groups,
andarise when there are vertices in the singular locus.
It can be shown that one can always pass to a finite cover that has no vertices but does have
non-empty singular locus (cf. the proof of Proposition 4.4 of [28]). On the other hand, it is shown(and used in a crucial way) in [28] that for certain orbifolds with vertices in the singular locus, then
the number of vertices can be increased without bound in passage to finite sheeted covers.
$Q$

$p$

$sing_{p}^{0}(Q)$$p$$\Sigma$$\Sigma$$A_{4},$$S_{4}$$A_{4},$$S_{4}$$A_{b}$$A_{6}$

2.5
Let be a non-cocompact Kleinian (resp. Fuchsian) group acting on Ill3 (raep.
) with finite
co-volume. Let
denote the subgroup of generated by parabolic elements of . Note thatis visibly a normal subgroup of , and we may define:$H^{2}$$\Gamma$$\Gamma$$\mathcal{U}(\Gamma)$$\Gamma$$\Gamma$$\mathcal{U}(\Gamma)$$V(\Gamma)=(\Gamma/\mathcal{U}(\Gamma))^{ab}\otimes z$

Q.

Setting $r(\Gamma)=\dim_{Q}(V(\Gamma))$ , then it follows Rom standard $argument8$ that
dimension of the space of non-peripheral homology.
Example 1: Since$PSL(2, Z)=P<(\begin{array}{ll}1 10 1\end{array})$,$(\begin{array}{ll}l 01 1\end{array})>$$\Gamma$meridians.

$L$is a link in$S^{3}$

with$S^{3}\backslash L=H^{3}/\Gamma$

denotes the, it $follow8$ that $r(PSL(2, Z))=0$ .

More generally if is a non-cocompact Fuchsian group such thatgenus 9, then $r(\Gamma)=2g$ .
Example 2: If$r(\Gamma)$$H^{2}/\Gamma$

has underlying space ofthen $r(\Gamma)=0$ , since$\Gamma$is generated by

2.6
Let be a prime, and let
denote the field of order . If $X$ is a group, space or orbifold, let $d_{p}(X)$be the dimension of $H_{1}(X;F_{p})$ With this we make the following definition that will be useful inwhat follows.
$p$$F_{p}$.

$p$

Deflnition: Let $X$ be a group, space or orbifold and let be a prime. Then a collecuonoffinite index subgroups or flnite-sheeted covers of $X$ unth index or degree [X : ] is said to havelinear growth of mod homology if$p$$\{X_{i}\}$$X_{1}$$p$$\inf_{i}\frac{d_{p}(X_{1})}{[X:X_{i}]}>0$.

Much of the discusslon in \S 4 is motivated by the extra structure that having non-empty singularlocus provides. An indicatlon of this is the following result proved in [26] (see also [28] in the caseof $p=2$ for a proof that does not use the Golod-Shafarevich inequality).

35

Theorem 2.2. Let be a compact orientable 3-orbifold uyith non-empty singular locus and a finitevolume hyperbolic structure. Let be a prime that dinides the order of an element of. Thenhas a tower of finite-sheeted covers $\{Q:\}$ that has linear growth of mod homology.
$Q$$\pi_{1}^{orb}(Q)$

$p$

$Q$

$p$

3

A geometric characterization of arithmeticity

A fundamental dichotomy of Margulis [40] (which holds much more generally) asserts that a Kleiniangroup of finite
is arithmetic if and only if the commens urator$c\triangleright volume$$\Gamma$Comm$(\Gamma)=${$x\in PSL(2,$

:

$C)$

$x\Gamma x^{-1}$is commensurable with$\Gamma$}is dense in $PSL(2, C)$ . $Mor\infty ver$, when is non-arithmetic the commensurator is also a Kleiniangroup of finite co-volume, which is the unique maximal element in the commensurability class of .
In this
we interpret this $g\infty metrically$. In particular we will prove:$\Gamma$$\Gamma$$8\bm{r}tion$

Theorem 3.1. Let $M=H^{3}/r$ be an orientable finite volume hyperbolic S-manifold. Thenarithmetic if and only if the follouting condition holds:Letbe a closed geodesic. Then there nists a finite sheeted coversuch that
admits an $0\dot{n}entation- poesen\dot{n}ng$ involution such that the fixed point set of image of 7.

$M\dot{u}$$\gamma\subset M$$M_{\gamma}arrow M$$\tau$$M_{\gamma}$$\tau$

Proof: That this condition is satisfi\’e when is arithmetic is prov\’e in [12] (and implicit in $|28]$ ).
We give thefor completenaes. We can assume without loss of generality thatderiv\’e$kom$ aquaternion algebra $B/k$ . Let
be ahyperbolic element whoee $axi_{8}A_{a}$ projects to .
Let
be chosenthat its. Now the Lie product ab–ba definae
is disjoint Rom
an lnvolution
for which theof rotation is the perpendicular bisector of.
inandDenote this $g\infty daeic$ by . As shown in [28] (sae Proposition 2.4), there is an order $O$ of $B$ for whichlies in the image in $PSL(2, C)$ of the normalizer of in B. This is an arithmetic Kleinian groupcommensurable with (sae [28] or [39] Chapter 6). Hence there is ahyperbolic element
whose
the
.
It follows thatis now the perpendicular bisector of the axae
. Repeating theandargument of the previous paragraph provides an involution fixing(namely arising $bom$ the Lieproduct of the elements and $aga^{-1}$ ) and liae in an arithmetic Kleinian groupcommensurablewith . To complete the proof, tak the core ofin , and let
be the corraeponding cover$\Gamma$$pr\infty f$$\Gamma i_{8}$$a\in\Gamma$$b\in\Gamma$$\gamma$$\alpha isA_{b}$$\epsilon o$$A_{a}$$\alpha is$$\tau_{a,b}$$A_{a}$$H^{3}$$A_{b}$$\delta$$O$$\tau_{a,b}$$\Gamma$$g\in\Gamma$$g\infty daeic\delta$$\dot{r}sA_{9}i\epsilon$$A_{a}$$aA_{9}$$A_{9}$$A_{a}$$g$$\Delta$$\Gamma$$\Gamma\cap\Delta$$\Delta$$M_{\gamma}$$ofM$

For the converse,
the manifold $M=H^{3}/\Gamma$ is non-arithmetic and satisfiae the $\infty ndition$ .
Then by the raeult of Margulis mention\’e above, Commthe unique $m\alpha imal$ element in thecommeourability $clas8$ of . We can assume without loss of generality that $\Gamma=Comm(\Gamma)$ (notethat we now allow the $exi_{8}tence$ of elements of Rnite order $\Gamma=Comm(\Gamma))$ .
Let
be acollection of hyperbolic elements in that lie in distinct cyclic
of up
to conjugacy in . We arethat the condition holds, andfor each we can construct
an involution that normahzae afinite index subgroup of and which has $axi8$ of rotation in
the axis ofLet
be the subgroups \infty otruct\’e above which arebySince
is commeIl\S urable with and is the unique maximal element it $foUows$ thatfor all .
Now contains only afinite number of cyclic groups of order 2up to conjugacy. Hence infinitely
many of thaee are conjugate in , and in particular there is a conjugacy thatthe aris ofto that of (say). Hence there is a conjugacy that takes the axis ofto that of afixed .
However, this contradicts how thaee elements were chosen.
$a\epsilon sume$$(\Gamma)ie$$\Gamma$$\{\delta_{\mathfrak{n}}\}$$subgrou_{\mathfrak{B}}$$\Gamma$$aesum\dot{\bm{o}}g$$\Gamma$$8O$$\Gamma$$\tau_{n}$$\delta_{n}.$$\Delta_{n}$$normali_{\mathbb{Z}}\text{\’{e}}$$\tau_{\mathfrak{n}}$$\Gamma$$\Gamma$$n$$\Gamma$$\tau_{n}\in\Gamma$$H^{3}$.

$\Delta_{n}$$n$$\Gamma$$\Gamma$$\tau_{n}$$\tau_{1}$$tak\infty$$\Gamma$$\Gamma$$\gamma_{n}$$\gamma_{1}$$0$A consequence of $Th\infty rem3.1$ is that every arithmetic Kleinian group is commensurable with one

containing an element of order 2. Indeed more is true, the foUowin is shown in [28].
$g$

36

Theorem 3.2. Let

group$\Gamma_{0}$such that$\Gamma$$\Gamma_{0}$be an arithmetic Kleinian group. Then is commensurable with a Kleinian
contains a group isomorphic to $Z/2ZxZ/2Z$ .
$\Gamma$

Remarks:(l) The $Z/2ZxZ/2Z$ constructed in Theorem 3.2 is related to a Hilbert Symbol forthe invarlant quaternion algebra. More precisely, if
is a Hilbert Symbol, then notice thatthe projection of the group generated by and to $PGL(2, C)\underline{\simeq}PSL(2, C)$ is simply a copy of$Z/2ZxZ/2Z$ .
$(\mathfrak{X})$

$i$

$j$

(2) In \S 7 we mention (see the proof of $Th\infty rem7.9$ ) another $g\infty metric$ characterization of Comm
In this case, it is the axes themselves that are used rather than the involutions.

4$(\Gamma)$.

Towards Conjectures 1.1-1.4 for arithmetic hyperbolic 3manifolds

Conjectures 1.1, 1.2, 1.3 and 1.4 still seem rather intractible for an arbitrary closed hyperbolic 3manifold. However, recent evidence points to the situation being somewhat better for those closed
hyperbohc 3-manifolds commensurable with a hyperbolic 3-orbifold with non-empty singular locus.
is evident in recent work of M. Lackenby, [26] and [27] as well as in [28]. A philosophicalreason that this $cue$ may be more amenable to study is that it is known from [11] that if $M$ isa finite volume non-compact hyperbolic 3-manifold, thenis large, so the strongest of theconjectures discussed in \S 1 holds in this settlng. Having non-trivial singular locus seems to be an
aid that helps replace the existence of the cusp (which is the crucial thing in the methods [11]) in
the cusped setting. As shown in \S 3, arithmetic Kleinian groups are always commensurable with$group_{8}$ containing non-trivial elements of finite order, and so fit into this picture.
On the other
hand, umost’ closed hyperbolic 3-manifolds are never commensurable with an orbifold as above.
$Thi\epsilon$$\pi_{1}(M)$

4.1
Here we sketch the proof of the following $th\infty rem$ .
Theorem 4.1. Conjecture 1. 2 implies Conjecture 1.4 for arithmetic hyperbolic S-manifolds.
This $th\infty rem$ wiu foUow immmediately from $Th\infty rem3.2$ and the next two results proved in [12]and [28] respectively. We can assume the manifold is closed by [11].
Theorem 4.2. Conjecture 1.2 implies Conjecture 1.

$S$

for closed arithmetic hyperbolic S-manifolds.

Theorem 4.3. Let $Q=H^{3}/\Gamma$ be a S-orbifold (utth possibly empty singular locus) commensurablewith a closed orientable hyperbolic S-orbifold that contains $Z/2ZxZ/2Z$ in its orbifold fundamentalgroup. Suppose that $vb_{1}(Q)\geq 4$ . Then is larye.
$\Gamma$

We will now sketch some of the ideas in the proofS of these results.

Sketch proof of Theorem 4.2:
Let $M$ be an arithmetic hyperbolic 3-manifold which is assumed to have positive flrst Betti number.
An important ingredient in the proof of $Th\infty rem4.2$ is the following general $g\infty metric$ result
proved in [12].

. Letbe a dosed hyperbolic 3-manifold. Then there is a dosed geodesic uith theproperty that it has a non-right angle transverse $inter\epsilon ection$ Utth every least area surface in $N$ .
$Th\infty rem4.4$$N$$\eta$

37

Given the geodesic
provided by the above theorem, $Th\infty rem3.1$ provides a finite cover
$M$
$M$
of
such that
admits an orientation-preserving involution such that the fixgd point set ofcontains a component of the preimage of . The key point now is to show that $b_{1}(M)>b_{1}(M)$ .
; $H_{2}(\overline{M})arrow H_{2}(M)$ is a rational
Suppose to the contrary they have the same rank, so thatisomorphism. Pick a connected embedded surface $F$ , whose homology class represents an eigenvalue
for the action of on $H_{2}(M)$ . $Con8ider$ the class $p_{*}[F]\in H_{2}(M)$ , this might not be primitive,
so take a least area embedded surface in $M$ representing the primitive class. Hence we may write$p_{*}[F]=a[G]$ for some integer . The following two lemmas are proved in [12].
$\eta\subset M$–$M$$\tau$$\tau$$\eta$$p$$\pm 1$.

$\tau_{l}$

$G$

$a$

Lemma 4.5. Let $p:\overline{M}arrow M$ be a finite sheeted covenng and suppose thathas the samerank as $H_{2}(M)$ .
Fix a connected embedded surface
in $M$ representing some nonzerv dass in $H_{2}(M)$ and let$H_{2}(\overline{M})$$F$$\overline{F_{1}},$be the components of $p^{-1}(F)$ .
Then for every $i,j,$in$\ldots,\overline{F_{k}}$$[\tilde{F}_{1}]=\pm[\overline{F_{j}}]$$H_{2}(\overline{M})$.

Lemma 4.6. In the notation of Lemma 4.5,
in addition that $F$ is least aoea for $[F]$ .
Then every
has the same aoea and this is least
.
class
for
Given thaee lemmas, the sktch of the proof Anish\’e as foUows. First, it can bethat
the surface
is connect\’e (otherwise $\pi_{1}(M)$ can be shown to be large). Now $p^{-1}(G)$ consistsof component\S each of which is an emb\’ed\’e $surfa\infty$ and therefore aprimitive $cla8S$ in.
Furthermore, since is arational isomorphism, $[F]=\pm[G^{\cdot}]$ , whereis any choice of acomponentof $p^{-1}(G)$ . It foUows we have that $\tau_{*}[G^{*}]=\pm[G^{t}]$ . $Mor\infty ver$, by Lemma 4.6, any suchis aleastarea surface in the homology.
By choice of , there is at least one component of the preimage of which $hu$ anonrightangle traoverse intersection with . Make this choice for. However, this surface can be us\’e toviolate aresult of Hass on intersrtioo of lea\S t area surfacae [21] to obtain a $\infty ntradiction$ . Briefly,
the surfacaeandare homologous up to orientation, leaet area, yet they meet withoutcoinciding for angle $rea8ons$ using the involution. Repeat\’e application of this increaees the first
Betti number without bound.
$\epsilon uppose$$\tilde{F}_{1}$$[\tilde{F_{1}}]$$\theta\iota e$$a\epsilon a$$ie$$ar_{1}\text{\’{e}}$$G$$H_{2}(\overline{M})$$G^{r}$$p_{n}$$G$$cla\epsilon\epsilon[G^{r}]=\pm[F]$$G$$\eta$$\eta\sim$$G^{\cdot}$$G^{*}$$\tau(G^{*})$$\square$Remarks:(l) An $intere8ting$ feature of this argument is that although it uses arithmetic in anaesential way, it is $g\infty metric$ , in that arithmeticity is used to produce involutions.
(2) $Th\infty rem4.2$ was proved in the case of congruence arithmetic hyperbolic 3-manifolds by Borelin [2]. This method of proof was recently generahzed to give different proo& of $Th\infty rem4.2$ in [1]and [53].

Sketch proof of Theorem 4.3:
Key to this proof is work of Lackenby, [25] and [26]. In particular the foUowing result of Lackenby
gives a method of proving largeness of an arbitrary finitely presented group (which is a consequenceof a stronger result in [25]).
Proposition 4.7. Let be a finitdy presented grvup, and let $\phi:Garrow Z$ be ahomomorphism. Let $G_{1}=\phi^{-1}(iZ)$ , and suppose that, for some prime
has linear growth of modhomology. Then is large.
$G$$su\dot{\eta}ective$$p,$$\{G_{1}\}$

$p$

$G$

The crucial hypothesis in Proposition 4.7 is the linear growth of mod homology. In the context
of orbifolds, as noted in \S 2, $Th\infty rem2.2$ provides a tower with linear growth in mod homology
for some prime . However, $Pr\dot{o}position4.7$ uses a very specific tower. This can also be established,
for in [28] it is shown that:
$p$

$p$

$p$

38

Proposition 4.8. Let be a compact orientable S-orbifold, and let $C$ be a component offor some prime . Let $p::|Q_{i}|arrow|Q|(i\in N)$ be distin finite sheeted covering spaces $of|Q|$ suchthat the restnction of to each component of
is a homeomorphism onto C. Let
be the$CO7ve$ sponding covering spaces of Q. Then
has linear growth of mod homology.
$Q$$sIng_{p}^{0}(Q)$

$ct$

$p$

$p_{i}^{-1}(C)$$p_{i}$$Q_{i}$$\{Q_{1}\}$$p$

Proof. A standard 3-manifold argument (see [28] Proposition 3.1) shows that$d_{p}(Q:)\geq|sing_{p}^{0}(Q_{i})|$By assumption$p_{j}^{-1}(C)$.

is a $hom\infty morphism$ onto $C$ and so it follows that$|sing_{p}^{0}(Q:)|\geq[Q_{j} :Hence $\inf_{i}d_{p}(Q_{i})/[Q:Q_{i}]\geq 1$ as required.

Q]$

.

$0$

It quickly follows from Proposition 4.7 and Proposition 4.8 that we have.
Corollary 4.9. Suppose
be a compact orientable -orbifold uith non-empty singular locus containing a circle component. Assume that that $b_{1}(Q)\geq 2$ , thenis larye.
$Q$$S$$\pi_{1}^{orb}(Q)$The proof of $Th\infty rem4.3$ is now completed as follows. By hypothesis,
has a finite coversuch that $b_{1}(Q’)\geq 4$ . Let $Q”$ be the hyperbolic orbifold, commensurable with , containing$Z/2ZxZ/2Z\ln$ its fundamental group. Now, $Q’$ and $Q”$ are commensurable, and hence they havea common cover $Q”’$ , say. Since $Q”’$ is hyperbolic, it has a manifold cover $M$ We may assume that
$M$ regularly covers $O”$ . Now,
does not decrease under finite covers, and so $b_{1}(M)\geq 4$ . Since$Marrow Q”$ is a regular cover, it has a group of covering transformations
Now,
contains $Z/2ZxZ/2Z$ , and henoe some singular point of “ has local groupthat contains $Z/2ZxZ/2Z$ . It follows that
contains $Z/2ZxZ/2Z$ . Letandbe thecommuting covering transformations of $M$ corresponding to the generators of $Z/2ZxZ/2Z$ . These
are involutions. Let be the composition of and , which also is an involution. For $i=1,2$
and 3, let
be the quotient. Since $h_{i}ha8$ non-empty fixed point set, sing is a non-empty
collection of simple closed curves with order 2.
It is shown in [28] that for at least one $i\in\{1,2,3\},$ $b_{1}(Q_{i})\geq 2$ . So, CoroUary 4.9 now showsis large for some , and hence so is .
$Q$

$Q’$

$Q$

.

$b_{1}$$G$$\pi_{1}^{orb}(Q’’)$

$Q$

$G$

$h_{3}$$h_{1}$$Q_{1}$$h_{1}$$h_{2}$$M/h_{\dot{l}}$$\pi_{1}^{orb}(Q_{i})$$i$$h_{2}$$(O_{i})$$\Gamma$$O$

4.2
Here we summarize what us known to us at this time about when Conjecture 1.4 holds for arithmetic
hyperbolic 3-manifolds $(U8ingTh\infty rem4.2)$ . We only consider closed manifolds, $8ince$ as mentioned
above, the finite volume non-compact case is completely understood by [11].
Throughout the discussion $M=H^{3}/\Gamma$ is an arithmetic hyperbolic 3-manifold with invaniant
quaternion algebra $B/k$ . Note that any proper subfield of is totally real.
$k$has a real subfleld of index 2: We distinguish two cases; when there are totally $g\infty desic$
surfaces and when there are not.
In general the existence of a totally $g\infty daeic$ surface in a closed hyperbolic 3-manifold allows
one to deduce that
holds (see [32] and [37]). In the case when $M$ is an arithmetic
hyperbolic 3-manifold, the existence of a totally $g\infty daeic$ surface forces $\infty nditions$ on $B/k$ . The
number field satisfies $[k:k\cap R]=2$ and has a Hilbert symbol of the form $(*)$ , where andare non-zero elements of $k\cap R$ (see [39] Chapter 9.5).
$k$$Co_{\dot{\mathfrak{U}}}ecture1.4$

$k$

$b$

$B$

$a$

39

If one considers the case when only $[k : k\cap R]=2$ , forgetting the condition on $B$ , it is also
known that $vb_{1}(\Gamma)>0$ . This is proved by various authors ([24], [31] and [38]).
In some of the cases when $[k:k\cap R]=2$ but no totally geodesic surface is present, there is also
$g\infty metric/topological$ proofs that $vb_{1}(\Gamma)>0$ . This happens when there is an orientation-reversing

involution on some manifold in the commensurability class of$H^{3}/\Gamma$.

Other conditions on subflelds: The paper [24] can be applied in some other cases as we now
describe.
Supposeis a proper subfleld and there is a tower of intermediate fields:$\ell\subset k$$\ell=\ell_{0}\subset\ell_{1}\subset\ldots\subset\ell_{m-1}\subset\ell_{m}=k$where each extension

is either a cyclic extension of prime degree or a non-Galois cubic
extension. Then $vb_{1}(\Gamma)>0$ .
In a similar spirit it is shown in [45] that if is contained inside a solvable Galois extension of, then $vb_{1}(\Gamma)>0$ . In particular a coroUary of these results is the following.
$\ell_{:}/\ell_{:-1}$$k$$\ell$

Corollary 4.10. Suppose $[k:Q]\leq 4$ , then$\Gamma$is large.

Proof: Any such extension has solvable Galois group over Q.

$O$

Clozel result: Perhaps the most general method known at present for proving Conjecture 1.2
for arithmetic hyperbolic 3-manifolds is [9]. In particular this applies to every field with onecomplex place, but there is a condition on the quaternion algebra. We state
result here in
the notation of this paper. Some topological consequences are discussed below.
$s$

$k$

$C1_{0\mathbb{Z}}e1’ s$be an arithmetic Kleinian grvup as above. Assume that for every place
contains no quadratic extension ofwhere is a rational prime and. ThenTheorem 4.11. Let$\nu\in RamrB,$$vb_{1}(\Gamma)>0$$k_{\nu}$.

$\Gamma$$Q_{p}$$\nu|p$$p$

43
There are $situation8$ when $Th\infty rem4.11$ can be used together with the existence of certain finite
subgroups in arithmetic Kleinian groups to prove Conjecture 1.4. For example, we have the foUowing
from [28].

Theorem 4.12. Let be an arithmetic Kleinian group commensurable with a Kleinian group containingoror a finite dihedral group that is derived jhvm a quaternion dgebra. Then is$\Gamma$$A_{4},$$S_{4}$$A_{6}$$\Gamma$large.

The proof of this foUows $bomTh\infty rem4.11$ on noticing that the existence of these finite subgroups places conditions on the invariant trace.field and quaternion algebras of the arithmetic
Kleinian groups (indeed this is true arithmetic or not). For example in the case oforthe$A_{4},$

invariant quaternion algebra is isomorphic to$( \frac{-1,-1}{k})$$S_{4}$$A_{6}$, and this can be shown to be unramifled at

all flnite places.
The conditions forced on the algebra are reminiscent of how the existence of a totaUy $g\infty daeic$
surface in arithmetic hyperbolic 3-manifold places conditions on the invariant tracefield and quaternion algebra (recall \S 4.2)

Remark: The discussions above, together with the discussion in \S 4.2, motivate the foUowing. This
provides a uniform setting for how the existence of a totally $g\infty daeic$ surface or a finite group of the40

has on the invariant quaternion algebra, and is perhaps more approachable thanform
orthe general problem.
$A_{4},$$A_{5}$$S_{4}$

be a field utth one complex place, and a (necessarily) totally real $su$ bfield. Supis an arithmetic Kleinian group urith invariant trace-field and invariant quatemion algebra

Challenge: Let

$p$

$k$

pose
that satisfies:$\Gamma$$k$$B$$B\cong A\otimes^{\ell}k$

for some quatemion algebra over$Br(\ell)arrow Br(k))$ . Show thatis large.
$A$$\ell(ie$.

$B$lies in the image of the natural map of Brauer groups$\Gamma$

Note also, that removing the hypothesis in $Th\infty rem4.12$ that the dihedral subgroup is derived
Rom a quaternion algebra would prove that all arithmetic Kleinian groups are large. It would beinteresting to give a $g\infty metric$ proof of $Th\infty rem4.12$ . Thus we pose as a warm-up challenge to thegeneral situation, and which should be more amenable.
Conjecture 4.13. Let.
is$\Gamma$be a Kleinian group offinite co-volume containing$A_{4},$$S_{4}$

or$A_{8}$. Then$\Gamma$$la\eta e$

Another interesting topological application of Clozel’s result is that it applies to show that if
is an arithmetic integral homology 3-sphere then $vb_{1}(\Sigma)>0$ , and henceis large. For in thiscase, the invariant trace-field of has even degree over $Q$ , and the invariant quaternion algebra ofis unramified at all finite places (see for example [39] $Th\infty rem6.4.3$ ). So $kom$ the perspective ofConjectures 1.1-1.4, in the arithmetic setting, integral homology spheres are easier to handle thanrational homology 3-spheres.
We close this subsection with two examples on the limitation of current techniques regarding the$\Sigma$$\pi_{1}(\Sigma)$$\Sigma$$\Sigma$arithmetic methods mentioned here.

Example 1: This is taken $kom[28]$ , and is acommensurability class of arithmetic 3-orbifolds forappliae to provide acover with positive first Betti number.
which none of the methods
We do this in the first possible degree, namely 5(sae Corollary 4.10).
Let $p(x)=x^{6}-x^{3}-2x^{2}+1$ . Then has three real roots and one pair of complex conjugateroots. Let be acomplex root and let $k=Q(t)$ . Now has one complex place and its Galois groupis . There is aunique prime of norm
in . It $foUow8$ thataquadratic exteoion of. Ibke $B$ ramified at the real emb\’edings and the prime . Then it is unknown whether any
arithmetic Kleinian group arising $komBha\epsilon$ acover with positive first Betti number.
Briefly, if is any group the commeourability clas8, then since has odd degree, the $discus8ion$in \S 4.2 shows that there are no non-elementary Rffiian subgroup8. The raeult of Clozel does not
apply by the condition on , and none of the other work discuss\’e above appliae since $[k:Q]=5$and the Galois group is
For this flnal part, note that if were contain\’e in asolvable exteoionof $Q$ , then the Galois $cl\infty ureK$ of would be asubfield of L. However, this impliae that the
Galois group $Ga1(K/k)$ is aquotient of $Ga1(L/k)$ , which i\S solvable, and this is acontradiction.
One can ako adjoin other primae to $Ram_{f}B$ that cootructs quaternion $algebra8$ for which
are torsion $hee$ .
$di_{8}cus\epsilon\text{\’{e}}$

$p$

$k$

$t$

$S_{6}$$11^{2}$$\mathcal{P}$$k$$k_{\mathcal{P}}i\epsilon$$Q_{11}$$\mathcal{P}$$\Gamma$$k$$\bm{i}$$S_{6}$$L$.

$\mathcal{P}$

$k$

$k$

$\Gamma_{O}^{1}$

In the next example, we construct a family of arithmetic orbifolds $Q_{p}=H^{3}/\Gamma_{p}(p$ a prime congruent
to 1 mod 4) where contains elements of order and for whichis unknown to be positive.
Note that the invariant tracefields of the orbifolds
have degree going to infinity (compare with
Corollary 5.6).
$\Gamma_{p}$$p$$vb_{1}(\Gamma_{p})$$Q_{p}$

Example 2: Let$p$be a prime congruent to 1 mod 4, and let $\ell_{p}=Q(\cos\pi/p)$ . Note that$\ell_{p}$contains

41

the quadratic extension$Q(\sqrt{p})$.

Claim:(l) There exists a number field$Ga1(k_{p}/\ell_{p})\cong S_{5}$$k_{p}$.

utth one complex place such that $[k_{p} : \ell_{p}]=5$ and(2) There exist infinitely many rationd primes
contains a quadratic extension of .

$q$such that if$\nu$

is a place of$k_{p}$

wzth$\nu|q$then$k_{P\nu}$$Q_{q}$

$(S)$

in

Let

$\zeta_{p}$denote a primitive p-th root of unity. Then, the primes$k_{p}(\zeta_{p})/k_{p}$

$\nu$

.

constructed in (2) do not split

Given these three conditions we can construct orbifolds as follows. Let and be as in (1) above,let
be a quaternion algebra ramifl\’e at all real places, and let be a place ofas in (2).
Since has even degree we insist thatis ramified at another place which we also assume is
as in (2). Since and do not split in, it follows that(see [39]embeds in
7.3.3 for example) and indeed, for any maximal order
the group
contains an elementof order (see for example [39] $Th\infty rem$ 12.5.4). Let.
As in Example 1 above, it is unknown using the methods described in this paper that $vb_{1}(Q_{p})>$. Briefly, although $[k_{p} : Q]$ has even degree there are no totally $g\infty doeic$ surfaces, since is the$m\alpha imal$
real subfield of which is of index 5. For if there were totally $g\infty desic$ surfaces, thenwould contain a totally real subfield of index 2 (see \S 4.3) which would contain and this is false.
As in Example 1, condition (1) above ensures that is not contained in any solvable extension of, and the condition on the prime ensures that Clozel’s condition fails.
$p$$B_{p}/k_{p}$$k_{p}$$k_{p}$$\nu$$k_{p}$$\nu’$$B_{p}$$\nu$$\nu’$$k_{p}(\zeta_{p})/k_{p}$$k_{p}(\zeta_{p})$$O_{p}\subset B_{p}$$p$$B_{p}$$\Gamma_{\mathcal{O}_{p}}^{1}$$Q_{p}=H^{3}/\Gamma_{Q_{p}}^{1}$$0$$\ell_{p}$$k_{p}$$k_{p}$$\ell_{p}$$k_{p}$$\ell_{p}$$\nu$Proof of Claim:

To prove (1), note that given the existence of a fieldwith $[k_{p} : \ell_{p}]=5$ and with one complex
place, thenor . This follows for example using the arguments of \S 3 of [8]. The
existence of such a field can be argued as follows. We seek to build a monic irreducible polynomial$p(x)=x^{6}+a_{1}x^{4}+a_{2}x^{3}+a_{3}x^{2}+a_{4}x+a_{6}(a:\in\ell_{p})$ , such that $p(x)$ has 1 pair of complex conjugateroots and 3 real roots, and furthermore, for every Galois embedding : $P_{p}arrow R$ the polynomial
obtained by applyinto the coefficients of has only real roots. Now
embeds in $R*^{-1}$as a dense subset, so that embeds in
as a dense subset. Hence a polynomial as above
gives a vector inand since the condition on the roots is an open condition it follows that
we can find a a polynomial satisfying the condition on the roots. $Mor\infty ver$ , by a similar argument,
the polynomial can be assumed irreducible.
To establish (2) and (3) we argue as follows. The Galois group of $Q(\zeta_{p})/Q$ is cyclic of order$p-1$ , so the Cebotarev Density $Th\infty rem$ gives infinitely many rational primes such that is
totaUy inert in(l.e. the inertial degree is precisely $p-1$ ). As remarked above,
contains, and so for any prime as above, if is the-prime lying over , and is a -prime
over , then contains. By properties of the inertial degree, $[Q(\sqrt{p})_{w} : Q_{q}]=2$ . This
proves (2).
Now (3) also follows, since choosing a prime as in (2), and a -prime lying over , thencannot split in. For if so, then it foUows that for the unique (from the previous paragraph)-prime with, that must split in, and this is false from above.
$k_{p}$$Ga1(k_{p}/\ell_{p})$$S_{6}$$\sigma$$p^{\sigma}(x)$$g\sigma$$p$$\ell_{p}$$R^{\lrcorner}l*-1\lrcorner$$p_{p}5$$p$$R^{f}\lrcorner b^{-\lrcorner 1}$

$q$

$q$

$Q(\zeta_{p})$$\ell_{p}$$Q(\sqrt{p})$

$q$

$q$

$Q(\sqrt{p})$

$w$

$q$

$k_{p}(\zeta_{p})$$\ell_{p}$

$\mu$

$q$

$\nu$

$k_{p}$$Q(\sqrt{p})_{w}$$k_{\nu}$$\nu|\mu$$\mu$$Q(\zeta_{p})$$\nu$$k_{p}$

$q$

$\nu$

42

4.4
The methods of [9], [31], [24], and [31] are those of automorphic forms and automorphic representations (see also [48] for a survey of some of this). In particular, these methods involve applying
some type of “functoriality”; eg Base Change, or the Jacquet-Langlands correspondence (see moreon this below). Perhaps the biggest chaMenge from the geometric or topological perspective is tounderstand $g\infty metric$ features of incompressible surfaces represented by duals to the cohomology
classes produced by these methods. In some cases, totally $g\infty daeic$ surfaces can be recognized in
the application of Base Change in quadratic extensison, but this is not always the case.
To complete the cussion about the connection betwaen the topoloy of finite covers of arithmetic
hyperbolic 3-manifolds and theof automorphic forms, we should mention that Conjecture
1.2 aconsequence of various conjectures in automorphic forms and the Langlands Program. One
such conjecture is mention\’e [9], heee we briefly discuss another that relates to eUiptic $curv\infty$ . Our
intention here not to give adetailed exposition of all the details, but simply to indicate to a
topologist, the gist of the connection, and the powerful implicatioo that theof automorphic
forms ha\S in theof arithmetic groups. Because we $wiU$ only give abrief discussion, we refer
the reader to [17] and [23] for more detaik regarding theof automorphic reproeentation\S .
We $empha\epsilon ize$ that what we discuss below is well-known to the experts in theof automorphic repraeentations, our purpoae merely to advertise the connection\S .
We begin with
Change for $GL(2)$ . Let be anumber field,
the adelae of and denote
by $A(k)$ the set of equivalenceof automorphic representations of $GL(2,A_{k})$ . Let
denotethe set of equivalence $clas8es$ of cuspidal automorphic repraeentations of $GL(2, A_{k})$ .
Suppose thatafiniteof number fields, then part of the Langlands Program
pr\’eicts that ifthen there is acanonicaUy associated $[\pi’]\in A(K),$ caU\’e the the Base
Change lifl of . Rrthermore if is cuspidal, then “typlcally”, is cuspidal. The existence of this$Ba\Re$ Changebaen $e8tabli_{8}hed$ in very few casae, and are the source of the $non- vani\epsilon hing$ ofin theof afield with one $compl\alpha$ place having certain proper subfiel&as discussed in \S 4.3.
For our purposae, we make the following assumption:$d\dot{w}$$th\infty ry$$i_{8}$$i_{8}$$th\infty ry$$\epsilon etting$$th\infty ry$$th\infty ry$$i_{8}$$Ba\epsilon e$

$k$

$k$

$A_{k}$$cla\epsilon 8\mathfrak{B}$$\mathcal{A}_{c}$$exten\epsilon ion$

$K/ki\epsilon$$[\pi]\in \mathcal{A}(k)$$\pi$$\pi’$$\pi$$1iRha\epsilon$$vb_{1}$$ca\epsilon e$

Assumption BC: Let
to
.

$k$

be a field with one complex place, then Base Change lifts$e\dot{m}t$fiom $A(Q)$$\mathcal{A}(k)$Given this we sktch the proof of:Theorem 4.14. Let $M$ be an arithmetic hyperbolic $S- manif_{0}u$ Under Assumption $BC,$ $vb_{1}(M)>$, and hence $\pi_{1}(M)$ is large.
$0$

Sketch Proof: We can assume that $M=H^{3}/\Gamma$ is closed. Denote the invariant trace-field ofby and $B$ respectively. Assume that $Ram_{f}B=\{\nu_{1}, \ldots, \nu_{l}\}$ . To show that $vb_{1}(M)>0$ , we $shaU$
construct a congruence subgroup of some groupin the commensurability class of and for which
the first Betti number is non-zero.
The connection with the cohomology is provided by Matsushima’s $th\infty rem[41]$ which says inthis setting, that if $X=H^{3}/\Delta$ is a closed orientable hyperbolic 3-manifold then $b_{1}(X)$ can becomputed as the multiplicity of a certain irreducible unitary representation of $SL(2,C)$ occuring in
the decomposition of $L^{2}(SL(2, C)/\Delta)$ under the right regular representation. We shall denote thisby
The goal is to show that the multiplicity of can be arranged to be non-zero by passage to acongruence subgroup. This is $ac$hiev\’e using Assumption BC together with the Jacquet-Langlands
Correspondence, the main result of which we state below in a somewhat abbreviated form (see [17]and [23]). To state this, we denote by
the adeles of the $divi_{8}ion$ algebra $B$ (see [39] Chapter
7 for example) andthe invertible adeles. Note that irreducible unitary representations of$\Gamma$$k$$\Gamma_{O}^{1}$$\pi_{0}$.

$\pi_{0}$$B_{A_{k}}$$B_{A_{k}}$$\Gamma$43

these adelic groups have the(the completed restricted tensor product), where
irreducible unitary representation of the local group at the place .
$form\otimes\pi_{v}$$\pi_{v}$

is an

$\nu$

Theorem 4.15. There is a correspondence which associates to each irreducible unitary representationof , (which is not l-dimensional) an imducible unitary representation of $GL(2, A_{k})$ .
Indeed, if
the corre spondence $\pi’arrow\pi$ is one-to-one onto the collection of cusp foms of$GL(2, A_{k})$ rvzth
a discrete series representation for each $\nu\in Ram$ B.
$\pi’$$B_{A}$$\pi$$\pi=\otimes\pi_{v}$$\pi_{\nu}$

The goal is to produoe acuspidal automorphic representation of $GL(2, A_{k})$ such that at theinfinite places $thi8repre8entationi\epsilon$ ofright type; $i.e$ has the form
(the $C$
factors $corre8ponding$ to the trivial representation of the invertible elements in the Hamiltonian
denote the primae a\epsilon sociat\’e to the flnite placae in$quaternions).Tothi_{8}end,1et’P_{1},.\cdots,\mathcal{P}_{l}$$\pi$$\pi_{0}\otimes C\otimes\ldots\otimes C$$\theta\iota e$$Ram\iota Banda\S 8umethatN\mathcal{P}_{j}=q_{j}=p_{J^{J}}^{t}$, let be an elliptic curve defined over with -invariant $1/N(s\infty[50]$Denoting byfor detaik about elliptic curvae). That such an elliptic curve
can be aeen explicitly $hom$ the$E$$N= \prod p_{j}$

$Q$

$j$

$\infty\dot{\alpha}st\S$following $con8truction$ on setting $j=1/N$ .
Let $j\in Q,$ $A=27j/(1728-j)$ and $B=2A$ . Then $y^{2}=x^{3}+Ax+B$ is an eUiptic curve defin\’eover $Q$ with -invariant . Now $E/Q$ is modular, and the level can be shown to be $NC$ for $wme$
integer C. In particular, modularity impliae that $E$ is attach\’e to acusp form of weight 2(which
is anewform) for, which in turn providae acuspidal automorphic $repr\infty entation\pi$ with$[\pi]\in A_{c}(Q)$ . We remark that the identification of this cuspidal automorphic $repr\infty entationkom$
the newform providae a cuspidal automorphic representation which is discrete seriae at the infinite
place of $Q$ and at those primae dividing $NC$ (sae [17]).
Assumption BC providae aBase Change $1iR\pi’$ , which is cuspidal in this $cue$ because of the$j- inva\dot{n}ant$ , which $guarant\infty s$ that the elliptic
curve $wiU$ have potentiaUy multiplicative r\’euction
at primae dividing $N$ over . It can also be shown that,at the infinite place of $Q,$ is discrete, then at the inflnite placae of this $repr\infty entationha\epsilon$ the right type. Furthermore, again
from the behavlor of , at thein $Ram_{f}B$ , and -primae lying over primae dividing $C$ , therepraeentation is necaesarily discrete seriae. $Th\infty rem4.15$ now provldae acuspidal automorphic$repre8entation$ for which
occurs at the complex place. Amore detailed analysis of the $corraeponrightarrow$dence actually shows that this provides acusp form on acongruence $8ubgroup$ of somewhere
the level $corraepond\epsilon$ to -primae lying over primes dividing $C$ and -primae lying over the primae
$j$

$j$

$\Gamma_{0}(NC)$$k$$\epsilon ince$$seri\infty$

$\pi$

$k$

$place\epsilon$

$\pi$

$k$

$\pi_{0}$$\Gamma_{O}^{1}$$k$$p_{j}$

.

$k$

$0$

4.5
We now discuss some other more general situations where the existence of orbifolds in a commensurabilty class are helpful. Firstly, there are other situations where the existence of certain
2-dimensional orbifolds can be used to prove Conjecture 1.1, 1.2, 1.3 and 1.4; eg as above whenthe 2-orbifold is totally $g\infty daeic$ by [32] (see also [38]). Here we indicate another cl , which again$provide8$ evidence that hyperbolic -manifolds commensurable with orbifolds do provide a morereaeonable clus to work with.
$\mathfrak{B}8$

Theorem 4.16. Let $Q=H^{3}/\Gamma$ be a closed hmperbolic 8-orbifold for whichis infinite.
Assume that contains an essential 2-orbifold with underlying space the 2-sphere or the Z-toms.
Suppose thata hyperbolic S-manifold commensurable utth , then $M$ is virtudly Haken.
$\pi_{1}(|Q|)$$Q$$M\dot{u}$$Q$

Proof: We can assume that is orientable, otherwise, by [34], has a finite sheeted
which isa non-orientable hyperbolic 3-manifold, and it is well known that such manifolds have positive first
Betti number (see [22] Chapter 6). Hence $vb_{1}(M)>0$ .
$Q$

$Q$

$\infty ver$

44

Letting $N$ denote the normal closure of all the elliptic elements of , then $\pi_{1}(|Q|)=\Gamma/N$ . Notethat the solution to the $G\infty metrization$ Conjecture ensures thatis residuaUy finite. Let:
be the quotient homomorphism.
We deal first with the case of the sphere. Thus suppose that $Sarrow Q$ is an emb\’ed\’e essential
2-8phere with cone point8. Slnce the orbifold group $F$ , of , is generated by eUiptic elements it
follows that $\phi(F)=1$ .
Now assume thatis atorsion hae subgroup of finite index in . Then $F_{0}=F\cap\Gamma_{0}$ is asurface group of genus at least 2. Restricting to
we have that $\phi(F_{0})=1$ . Hence, since
is infinite, and raeiduaUy finite, we deduce that $F$ is contained in infinitely many subgroups of finite
contain\’e in infinitely many subgroups of finite index inindex in , and henoe. It is araeult of Jaco (sae [49] Cordlary 2.3) that, and therefore $M$ , is virtually Hakn.
The $ar_{1}ment$ in the case when is the 2-torus with cone points is handled in asimilar way.
However, here note that $\phi(F)$ ne\’e not be trivial, but it is aquotient of $Zx$ Z. Thus $\phi(F)$ is an
Abelian subgroup ofof rank at most 2. This is either trivial or isomorphic to one of $Z,$,
$Z/nZ$ or $ZxZ/2Z$ (see [22] Chapter 9). If $\phi(F)$ is trivial we argue as above. Thus assume that
$A=\phi(F)$ is one of the non-trival Abelian groups listed above. The case of $A=Z\oplus Z/2Z$ is
ruled
out since the existence of sui asubgroup forces $|Q|$ to be non-orientable (soe [22] $Th\infty rem9.12$ )contrary to the $as8umption$ .
SuPpose first that
irreducible. Note that has inflnite index in, and since
is infinite, it follows $kom[22]$ Chapter 9that is isomorphic to or. In thaee casae[20] shows that is aseparable subgroup of(aesuming the $G\infty metrization$ conjecture). We
argue
now
as above using the raeult ofIf
not irreducible, then either $|Q|=S^{2}xS^{1}$ or $|Q|$ decomposes as anon-trivial connect
sum. In the former case we obtain asurjection onto , and we may dduce that $M$ is virtually H&nthis. In the latter caae, the $ar_{1^{ment}}$ above will deal with all cases other than $A=Z/nZ$ .
However, in this case, [22] $Th\infty rem9.8$ givae adecomposition $\pi_{1}(|Q|)=\pi(X)*G$ , where $X$ is aclos\’e 3-manifold, $|G|<\infty$ andconjugate into G. Now
infinite, so $\pi_{1}(X)\neq 1$ . Hencesurjects anon-trivial hae product, and so this determines anon-trivial action of on atree. It
follow8 that $\pi_{1}(M)$ must contain afinite index subgroup that admits anon-trivial action on atrae,
virtually Haken.
and so$\Gamma$$\pi_{1}(|Q|)$$\phi$$\Gammaarrow\Gamma/N$$S$$\Gamma_{0}$$\Gamma$$\phi$$\Gamma$$F_{0}$$\pi_{1}(|Q|)$$F_{0}i_{8}$$\Gamma_{0}$$H^{3}/\Gamma_{0}$$S$$\pi_{1}(|Q|)$$Z^{2}$$\pi_{1}(|Q|)i_{8}$$A$$\pi_{1}(|Q|)$$\pi_{1}(|Q|)$

$A$

$A$

$Ja\infty$.

$Z$$Z^{2}$$\pi_{1}(|Q|)$$|Q|i\epsilon$$Z$$\hslash om$$A$$i\epsilon$$\pi_{1}(|Q|)i\epsilon$$\Gamma$$\Gamma$$Mi\epsilon$$0$

We remark that it is shown in [28] that any arithmetic hyperbolic 3-orbifold always admits a finite
cover which is also an orbifold with non-trivial singular locus and satisfies the condition on thefundamental group of the underlying space given in $Th\infty rem4.16$ .

A well-known approach to proving Conjecture 1.1 is to show that a closed hyperbolic 3-manifold
has a finite sheeted cover for which the character variety (of $(P)SL(2,C)$ representations) contains
a positive dimensional component. For a closed hyperbolic 3-manifold even constructing “new”
characters in a finite sheeted cover seems hard. We now address this issue exploiting orbifolds.
Note that if $M=H^{3}/r$ with $k=Q(tr\Gamma)$ , then any Galois embedding of defines a character
obtained by $appl\dot{p}g\sigma$ to the character of a faithful discrete representation.
$\sigma$$k$$\chi_{\sigma}$

Deflnition: Let $M=H^{3}/\Gamma$ be a closed orientable hyperbolic S-manifold. Say that $M$ has extra
characters if there eststs an infinite irreducible $(P)SL(2, C)$ representation of that is different
from for any Gdois embedding of $Q(tr\Gamma)$ .
In addition we shall say that $M$ virtually has extra characters if contains a finite index subgroupsuch thathas extra characters.
$\Gamma$$\chi_{\sigma}$$\Gamma$$\Delta$$H^{3}/\Delta$.

Let $M=H^{3}/r$ be a dosed onentable hyperbolic S-manifold commensurable urith
an orb 珂 old commensurable that contains $Z/2ZxZ/2Z$ in its orbifold fundamentd group. Then$Th\infty rem4.17$

45

either $vb_{1}(M)>0$ or$M\dot{m}\hslash ually$

has extra characters.

Proof: $Th\infty rem8.1$ of [28] shows that, under the hypothesis of $Th\infty rem4.17,$ $M$ is commensurablewith an orbifold for whichis infinite. It foUows thata finite index subgroup suchthat
surjects
where $X$ is a closed orientable 3-manifold with
infinite. Assuming
the $G\infty metrization$ Conjecture, we can usume that $X$ is hyperbolic, otherwise it is easily seen that$vb_{1}(X)>0$ and so $vb_{1}(H^{3}/\Gamma_{0})>0$.
When $X$ is hyperbolic, the faithful discrete representation ofprovides an extra character(it is different frombecause this representation is not faithful on ).
$Q$$\Gamma hu$$\pi_{1}(|Q|)$$\Gamma_{0}$$\pi_{1}(X)$$\Gamma_{0}$$\pi_{1}(X)$$\pi_{1}(X)$$\Gamma_{0}$$\chi_{\sigma}$$\square$Together with $Th\infty rem4.1$ we have.

Corollary 4.18. Let $M$ be an arithmetic hyperbolic S-manifold. Then either $\pi_{1}(M)$ is
virtually has extra chamcters.

$la\eta e$

or $M$

This also gives.
Corollary 4.19. Let $M$ be an
hyperbolic 8-man 珂 old. Then $M$ has a finite sheeted cover$N$
for which the character variety of $N$ contains at least one more component of characters ofirreducible representations.
$at\dot{\tau}\theta\iota me\hslash c$

Proof: By Corollary 4.18 either $\pi_{1}(M)$ is large, so that $\pi_{1}(M)\infty ntaio$ a finite index subgroupsurjecting a free non-abelian group or there is a finite index subgroup with an extra character. Bothcaees provide extra components in the $(P)SL(2, C)$ character variety.
$\square$4.6
Another well-known conjecture in the same spirit as Conjecture 1.1-1.4 is the virtual fibering conjecture of Thurston.
Conjecture 4.20. Let $M$ be a finite volume hyperbolic 3-manifold. Then $M$ has a finite sheeted
cover that fibers over$S^{1}$.

Unlike Conjectures 1.1-1.4 where there is a considerable amount of supporting evidenoe (even in
the arithmetic case), little is known about Conjecture 4.20. For the complements of knots andthere is work of [4], [15], [30], and [57] that provides some evidence, but for closed manifolds almost
nothing is known. However, in the arithmetic setting, many well-known examples are known to be
virtually fibered (see [5] and [47]). Arithmeticity is used in that the pairis a complete
invariant of the commensurabilhty clus, and so one can construct bundles (for example by surgery
on fibered knot complement8) and just $\infty mpare$ arithmetic data (see [5] and [47] for more details).
$li\grave{n}ks$$(k\Gamma,A\Gamma)$

5

The 2-dimensional situation

In this section we address the following question. By saying a Fhschsian groupmean that the underlying spaoe of the orbifold
hae genus .
$H^{2}/\Gamma$$\Gamma\dot{u}$of genus , we$\theta$$0$

Question 5.1. Are there infinitely many commensurability dasses of arithmetic $R cMan$ groupsof genus ?
$\theta$

46

The motivation for this question is the situation in dimension 3 (discussed in \S 6). Note that if
is a cocompact Fuchsian group of genus , thenis finite, and so the orbifold
is a “rational
homology 2-sphere”. Conjectures 1.2-1.4 pr\’eict that on paesage to certain finite covers, arithmetic
hyperbolic 3-manifolds stop being rational homology 3-spheres. Thus it seems interesting to discuss
how “widely distributed” arithmetic rational homology 3-spheres are. This is address in \S 6 when$\Gamma$$\Gamma^{ab}$$0$$H^{2}/\Gamma$

we discuss an analogue of Question 5.1 for arithmetic rational homology 3-spheres.
The answer to Question 5.1 is provided by the following result from [36].
Theorem 5.2. The answer to Question 5.1

no.

$\dot{u}$In contrut, it is easy to construct infinitely many cocompact arithmetic Fuchsian groups ofgenus (see [36] Corollary 4.10). Before discussing the
we recaU some notation and discuss aclaesical situation which $Th\infty rem5.2$ generalizes.
$0$$pr\infty f$

51
RecaU that a congruence subgrvup of $PSL(2, Z)$ is any subgroup of $PSL(2, Z)$ that contains a principal
congruence subgroupof level for some where$\Gamma(n)$

$n$

$n$

$\Gamma(n)=P\{A\in SL(2,Z):A\equiv Imod n\}$.

More generally, a non-cocompact arithmetic IFMchsian group is congruenoe if some conjugate ofin $PGL(2,R)$ contains some. Any non-cocompact arithmetic Fuchsian group is conjugate intoa maximal group commensurable with $PSL(2, Z)$ , and these can be described as foUows.
Any Eichler order of $M(2, Q)$ (an interesection of two maximal orders) is conjugate to$\Gamma$$\Gamma$$\Gamma(n)$$\mathcal{E}_{0}(n)$

$:=$

{

$(\begin{array}{ll}a bc d\end{array})\in M_{2}(Z)|c\equiv 0$

mod

$n$

}

and the conjugacy claeses of the maximal Fuchsian group8 we the groups
for square-free(the images in $PSL(2,R)$ of the normalizers of the Eichler orders). These groups have a morefamiliar description ae the normalizers in $PSL(2,R)$ of the groups$\Gamma_{\mathcal{E}_{0}(n)}^{+}$$n$$\mathcal{E}_{0}(n)$$\Gamma_{0}(n)=P${$(\begin{array}{ll}a bc d\end{array})$:$c\equiv 0$

mod

$n$

}.

In particular these maximal arithmetic Fuchsian groups contain, and so are congruenoe subgroups.
In the context of cusped 2-orbifolds, the analogue of a rational homology 2-sphere ls when $r(Q)=$(recaU \S 2.5). Thus before studying the cocompact setting, an eaeier question is to study noncocompact arithmetic Fuchsian group8 with $r(\Gamma)=0$ (there is after all only one commensurability$\Gamma(n)$$0$$\Gamma$claes).

Now it is well known that $r(\Gamma_{0}(n))=0$ if and only if $n\in\{1, \ldots, 10,12,13,16,18,25\}$ , and the sizeofcarries important information regarding theof eUiptic curves over Q. Furthermore itis known when
and that this is also related to interesting number $th\infty oetic$ phenomena;for example it is known that there are only finitely many such that, and that thesevalues of are related to the properties of the Monster simple group. Indeed, Ogg (cf. [43]) observedthat that those primes for which the group8
have genus , coincides with those primes
dividing the order of the Monster simple group (see [10], [36] and [43] for more details).
On the other hand, it is eaey to construct infinitely many subgroupsof finite index in$PSL(2, Z)$ for which $r(\Gamma_{t})=0$ ; e.g. by constructing certain cyclic covers of the the thrice-punctured
sphere.
$th\infty ry$$r(\Gamma)$$r(\Gamma_{8_{0}(\mathfrak{n})}^{+})=0$$n$$r(\Gamma_{\mathcal{E}_{O}(\mathfrak{n})}^{+})=0$

$n$

$p$

$N(\Gamma_{0}(p))$

$0$

$p$

$\{\Gamma_{i}\}$

47

52
The description of maximal arithmetic Fuchsian groups and congruence subgroups in terms of Eichler
orders can be extended to the cocompact setting which we briefly recall (see [36]).
Let $B$ be a quaternion algebra over a totally $re$ al fleld and let be a maximal order of . Let
be an integral 2-sided -ideal in which means that . is a complete -lattice in $B$ such that
$O$

$k$

$I$

$B$

$\mathcal{O}$$R_{k}$$O=\{x\in B|xI\subset I\}=\{x\in B|Ix\subset I\}$

The principal congruenoe subgroup of$\Gamma_{\mathcal{O}}^{1}$of level , denoted$O^{1}(I)=$

Since

$O/I$

is a finite ring,$O^{1}(I)$

$B$

$I$

$I$

{

$\alpha\in O^{1}$

hae finite index in: a–l$O^{1}$$\Gamma_{O}^{1}(I)$$\in I$.

is the image in $PSL(2,R)$ of}.

.

Deflnition: Let be an arimmetic hchsian group.
is cdled a congruence subgroup if there isa quatemion algebra defined over the totally real field ramified at all real places except one,
maximd order and an ideal I of as described above such that contains.
$\Gamma$$\Gamma$

$B$

$k$

$O$

$a$

$\Gamma$$\mathcal{O}$$\Gamma_{O}^{1}(I)$As in the caee of the modular group, any maximal withmetic IFMchsian group is a congruencesubgroup (see [36]). In addition to proving Thmrem 5.2, the methods also provide a generalizationof the results when $r(\Gamma)=0$ for a maximal (or congruence) arithmetic Fuchsian group (see [36]).
$\Gamma$

Theorem 5.3. There are finitely many conjugacy classes of congruence arithmetic $R\iota chsian$ groupsof genus zero. There are thus finitdy many conjugacy dasses of maximd arithmeM $h$ chsian gmups

of genus .
$\theta$

5.3
The proof of $Th\infty rem5.2$ is completed in the foUowing steps.

1. If a commensurability clus contains an arithmetic Fuchsian group of genus zero, then any
Fuchsian group containing will also have genus zero, so we can usume that we have an infinite
sequenoeof non-conjugate maximal arithmetic Fuchsian group8 of genus .
, sinoe ae noted in \S 2.3, there are only finitely many $\infty njugacy$
Note that Area
classes of arithmetic Fuchsian groups of bounded area.
$\Gamma$$\Gamma$$\{r_{:}\}$$0$$(H^{2}/\Gamma_{i})arrow\infty$

2. We now exploit the following results that give contrasting behavior of the first non-zero eigenvalueof the Laplacian. The first is an important result about the first non-zero eigenvalue of the Laplacianfor congruence arithmetic $Hl\iota chsian$ groups (see [18], [23] and [55]).

Theorem 5.4. If$\Gamma$

is a congruence arithmetic hchsian group then$\lambda_{1}(\Gamma)\geq\frac{3}{16}$where$\lambda_{1}(\Gamma)$,

is the first non-zero eigenvalue of the Laplacian of$\Gamma$.

Note by the discussion in \S 5.2, this applies to the maximal withmetic Fuchsian groups
the other hand, we have the following result of P. Zograf [58].
$Th\infty rem5.5$$g(\Gamma)$. If Area. Let be a thchsian group of finite co-area and let the genus of$\Gamma$$(H^{2}/\Gamma)\geq 32\pi(g(\Gamma)+1)$, then$\lambda_{1}(\Gamma)<\frac{8\pi(g(\Gamma)+1)}{Area(H^{2}/\Gamma)}$.

$H^{2}/\Gamma$$\Gamma_{1}$. On

be denoted by

48

3. We now have a contradiction, for with $g(\Gamma_{i})=0$ , and Area
that
. However this contradicts $Th\infty rem5.4$ .
$\lambda_{1}arrow 0$$(H^{2}/\Gamma_{i})arrow\infty$, Theorem 5.5 implies$\square$Corollary 5.6. There is an upper bound to the order of an dliptic element in an arithmetic hchsiangroup of genus .
$0$

Proof: Sinoe there are only finitely many commensurability classes of arithmetic IFUchsian groupsof genu80, there is a bound on the degree of the defining fields of these groups. If an arithmetic
Fuchsian group contains an element of order $n\geq 2$ , then contains $Q(\cos 2\pi/n)$ and it is well-known
that the degree of this fleld goes to infinity with .
$k$

$k$

$n$

$\square$

Computations sugge8t (see [36]) the following conjecture.
Conjecture 5.7. If
degree at most 7.

$\Gamma$

is an arithmetic $P4chsian$ grouP of genus , then the invariant tmce-field has$\theta$

Remarks:(l) The inequalities in $Th\infty rems5.4$ and 5.5 give an upper bound on the co-area of a
maximal (resp. congruence) arithmetic Fuchsian group of genus zero $u$ :Area$( H^{2}/\Gamma)\leq\frac{128}{3}\pi$(2) The same arguments hold for wbitrary fix\’e .
(3) Another feature of genus zero 2-orbifolds is that it is shown in [36] $Th\infty rem1.5$ , that any Fuchsian group of genus hae a global upper bound to the length of the shortest translation length of a
hyperbolic element.
$g$

$0$

6

Arithmetic rational homology 3-spheres

As discussed in \S 4, the fact that arithmetic hyperbolic 3-manifolds we commensurable with a 3orbifold affords $g\infty metric$ and topological approaches to the Conjectures 1.1-1.4. The discussion of
the caee of 2-dimensional rational homology $8pherae$ in \S 5 motivates some interesting questions indimension 3, that relate to Conjectures 1.1-1.4 but in the “opposite direction”.
The analogue of Question 5.1 is:
Question 6.1. Are there infinitely many commensurability classes of arithmetic rationd homology
S-spheres (even $dlo$ wing for orbifolds)?
Continuing with the analogy, another $\infty ns\Re uence$ of $Th\infty rem5.2$ is that there is an upperbound on the degree of the invwiant tracefield of an arithmetic Fuchsian group of genus (fromthe area bound described in \S 5.3). Henoe we pose:$0$

Question 6.2. Is there an upper bound to the degree ofrationd homology S-sphere?

$\theta\iota e$invariant trace-fidd of an arithmetic

Note that a positive angwer to Question 6.2 need not imply a negative answer to Question 6.1.

49

61
Here we provide a conjectural aPproach taken from [36], that provides a positive answer to Question
6.1.
As in the 2-dimensional $cue$ , we stwt with the non-cocompact setting which again is closely
related to the theory of elliptic curves over quadratic imaginwy number fields.
Let be a square-free positive integer, andbe the ring of integers in $Q(\sqrt{-}Z|)$ . The anb
logue of the modular group here is a family of groups, The Bianchi groups, $PSL$ (2, Od). In thissetting, The Cuspidal Cohomology Problem posed in the $1980’ su$ ked which Bianchi group8 have$r(PSL(2, O_{d}))=0$ . In [56] it wae shown that:$d$$O_{d}$

Theorem 6.3. $r(PSL(2, O_{d}))=0$ if and only if$d\in \mathcal{D}=\{1,2,3,5,6,7,11,15,19,23,31,39,47,71\}$.

As in dimension 2, there is a clus of congruence subgroups which play a key role in connectingwith the arithmetic. For an ideal
we define the congruenoe subgroup:$A\subset O_{d}$$\Gamma_{0}(\mathcal{A})=P${$(\begin{array}{ll}a bc d\end{array})$:$c\equiv 0$

mod $A$ }Set. Unlike the situation in dimension 2, $r(A)$ is far from understood atpresent. On the other hand, as in dimension 2, it is eaey to construct infinitely many examples of$non- co\infty mpact$ arithmetic Kleinian groups
for which $r(\Gamma)=0$ ; e.g. there are infinitely arithmetic
links inwhose complements are arithmetic.
$r(\mathcal{A})=r(\Gamma_{0}(\mathcal{A}))$$\Gamma$$S^{3}$In the context of the groups
suggest the following:$Co$njecture 6.4. For each$\Gamma_{0}(\mathcal{A})$$d\in \mathcal{D}$work of Grunewald and Mennicke [19], and Cremona [14],, there$\dot{\varpi}st$infinitdy many ideals$A\subset O_{d}$such that $r(A)=0$ .

For the caee of $Z[i]=O_{1}$ , more precise statements can be made. For example, the foUowing seems
reaeonable.

$Co\Phi ecture6.5$.

There extst infinitely many pairs of prime ideals
such that$(b)$ In fact, let $\mathcal{P}=<1+i>$ , then there are infinitely many prime
idealsmod 12 and
.

$(a)$

$\{P_{1},\mathcal{P}_{2}\}\subset Z[i]$$r(\mathcal{P}_{1}\mathcal{P}_{2})=0$$\mathcal{P}_{j}\subset Z[i]$

$1$

with

.
$N\mathcal{P}_{j}=$$r(\mathcal{P}\mathcal{P}_{j})=0$

As evidence of this, it is shown in [14] that of all integral ideals
with norm $\leq 5N$ only
76 give rise to groups with
. With reference to Conjecture 6.5(b) [14] shows that this holds
for prime ideals of norms 13, 37, 61, and 73, but fails for a prime of norm 97.
The relevanoe of these conjectures to the cocompact setting foUows from the Jacquet-Langlands
correspondence (recaU $Th\infty rem4.15$ ). Aof this is (cf. [6] $Th\infty rem3.3$ ).
$A\subset Z[i]$$r(\mathcal{A})>0$$\infty n\epsilon equence$$Th\infty rem66$. Let

mmiffing $B$ being,
then

$B$

be a division dgebra of quatemions over $Q(\sqrt{-d})$ with th set of p ヴイ mes. Let be a maimal order of , and. If is infinite,$\{\mathcal{P}_{1}, \ldots,\mathcal{P}_{2r}\}$$r(\mathcal{P}_{1}\ldots \mathcal{P}_{2r})\neq 0$

.

$e$

$O$

$B$

$\Gamma=\Gamma_{O}^{1}$$\Gamma^{ab}$A coroUary of this result and Conjecture 6.5(b) is the following. We remark that the primein Conjecture 6.5(b) is chosen so that the resultant groups
are torsion-free.
$\Gamma_{O}^{1}$

Corollary 6.7. If Conjecture $6.5(b)$ holds, then Question 6.1 has a posifive answer.

$\mathcal{P}$50

We finish this subsection with some further discussion motivated from the 2-dimensional setting.
Corollary 5.6 providae an upper bound, say , to the order of
element in an arithmetic Fuchsian group of genus . Rephraeing this in a way that (although vacuous in dimension 2) is suggestivefor dimension 3, Corollary 5.6 shows that if an arithmetic Fuchsian group is commensurable witha group containing an element of order $>N$ then is large. This Prompts in the spirit of \S 4 asituation that may be more approachable than the general $cue-this$ is very much in the spirit ofsome of the results in [27].
$N$$eU\ddagger ptic$$0$$\Gamma$$\Gamma$Show that for $dl$ sufficiendy large integers $N$ , any $(a|\dot{\tau}thmetic)$ Kleinian grvup commensurable rvith
one containing an element of order $N$ is large.

Recall that if$M=H^{3}/\Gamma i\epsilon$

acloeed orientable hyperbolic 3-manifold, the $injecti\dot{w}ty$ mdius of $M$(denot\’e by injrad $(M)$ ) is defined $u$ the largaet number sui that for all $p\in M$ , the ball $B(p,r)$
center\’e at of radiu8 is isometric to the -ball in. D. Cooper uk\’e whether there is a $K\in R$ ,independent of $M,$ such that injrad $(M)\leq K$ ?
Thi8 quaetion is of interest since, by raeidual $finiten\infty$ , an affirmative answer to thi8 quaetion,
impliae every closed hyperbolic 3-manifold $hu$ acover with flrst betti number at leaet 1. However,F. Calegwi and N. Dunfield [6] aower\’e thi8 quaetion in the negative; unhke the 2-dimensional ca\S e
$r$

$P$

$r$

$r$

$H^{S}$$discu88ed$ in \S 5.3.

In ], the author8 $co$ nstruct arithmetic rational homology 3-sphere8 arising $hom$ the division
algebra $Q(\sqrt{-2})$ ramified at the two places lying above 3in $Q(\sqrt{-})$ and with wbitrarily large
radii. The cootruction of [6] assumae the existence ofGalois repraeentatioo of$Ga1(\overline{Q(\sqrt{-2})}/Q(\sqrt{-2}))$althoughpr\’eicted
thatby the general ffamework of the Langlands corHowever an unconditional proof of the roeult of [6] wae
raePondenoe are not yet known toaetablished in [3]. It is worth remarking in light of $Qu\infty tion6.1$ that the construction of [6] providae
commensurable rational homoloy 3-sphere8. Ako these examplae show that there are infinitely$|6$$inj\bm{r}ti\dot{\mathfrak{n}}ty$$certa\dot{\bm{o}}$$exi_{8}t$.

many arithmetic rational homology 3-sPherae in the same commeourability $clas8C8(u$ in the 2dimensional setting with orbifolds).
Another interaeting feature of the cootruction of [6] is that their manifolds are all Haken. Thu8
it saems interesting to refine Cooper’8 orginal quaetion to.

Question 6.8. Are there non-Haken hyperbolic S-manifolds of arbitrarily larye injectivity radius(even alloutng for orbifolds)?

6.2
There are many examples of arithmetic rational homology -spheres. These can be found efficiently
u8ing the census of hyperbolic 3-manifolds developed using the computer program8 SnapPea, andthe exact version Snap (see [13]). We list the 108mallest known arithmetic hyperbolic 3-manifolds.
The first two we known to be the smallest two by [7]. We list these using the notation of [13].
) $3(-3,1)$ ; Volume 0.94270736277692772092.
1: ManifOld
Minimum polynomial: $x^{3}-x^{2}+1$ ; Root: .
Discriminant: $-23_{j}$ Signature $(1, 1)$ .
Real Ramification [1]; Finite Ramification .
Integral Traces; Arithmetic.
$m\alpha$$- 2$$\nu_{5}$2: Manifold $m003(-2,3)$ ; Volume 0.9813688288922320880914.
Minimum polynomial: $x^{4}-x-1$ ; Uot: 3.
Discriminant: $-283$ ; Signature $(2, 1)$.

51

Real Ramification $[1, 2]$ ; Finite Ramification
Integral Ttaces; Arithmetic.

$\emptyset$.

3: Manifold $mOlO(-1,2)$ ; Volume 1.01494160640965362502120.
Minimum polynomial: $x^{2}-x+1$ ; Root: 1.
Discriminant: $-3$ ; Signature $(0,1)$ .
Real Ramification ; Finite Ramification.
Integral ‘baces; Arithmetic.
$\emptyset$$\nu_{2},\nu_{3}$4: Manifold $m\infty 3(-4,3)$ ; Volume 1.263709238658043655884716.
Minimum polynomial: $x^{4}-x^{3}+x^{2}+x-1;$ Root: .
Discriminant: $-331$ ; Signature $(2, 1)$ .
Real Ramification $[1, 2]$ ; Finite Ramification .
Integral Tracae; Arithmetic.
$- 3$$\emptyset$

5: Manifold $m004(6,1)$ ; Volume $1.28u8530046835$ 44424603370.
Minimum polynomial: $x^{3}+2x-1$ ; Root: 2.
Discriminant: $-59$ ; Signature $(1, 1)$ .
Real Ramification [1]; Finite Ramification .
Integral Traces; Arithmetic.
$\nu_{2}$6: Manifold $m\infty 3(-3,4)$ ; Volume 1.414061044165391581381949.
Minimum polynomial: $x^{3}-x^{2}+1$ ; Root: 2.
Discriminant: $-23$ ; Signature $(1, 1)$ .
Real Ramification [1]; Finite Ramification.
Integral Naees; Arithmetic.
$\nu_{19}$7: Manifold $mW3(-5,3)$ ; Volume 1.54356891147185507432847.
Minimum polynomial: $x^{6}-x^{3}-2x^{2}+1_{j}$ Root: 4.
Discriminant: $-4511$ ; Signature $(3, 1)$ .
Real Ramification [1, 2, 3]; Finite Ramification.
Integral Traces; Arithmetic.
$\nu_{13}$8: Manifold $m\infty 7(4,1)$ ; Volume 1.583166660624812836166028.
Minimum polynomial: $x^{3}+x-1$ ; Root: 2.
Discrininant: $-31$ ; Signature $(1, 1)$ .
Real Ramification [lj; Finite Ramification.
Integral Traces; Arithmetic.
$\nu_{13}$9: ManifOld $m\infty 6(3,1)$ ; Volume 1.588646639300162988176913.
Minimum polynomial: $x^{3}-x^{2}+x+1$ ; Hoot: .
Discriminant: $-44$ ; Signature $(1, 1)$ .
Real Ramiflcation [1]; Finite Ramification .
Integral Traces; Arithmetic.
$- 2$$\nu_{2}$10: Manifold $m015(5,1)$ ; Volume 1.7571260291884513628747465.
Minimum polynomial; $x^{5}-x^{4}-x^{3}+2x^{2}-x-1$ ; Root: .
Discriminant: -4903; Signature $(3, 1)$ .
Real Ramification [1, 2, 3]; Finite Ramification.
$- 4$$\nu_{13}$52Integral Traces; Arithmetic.

We also give an example of an arithmetic rational homology 3-sphere that is of the type prdictedby Theorem 6.6.
Example: Let $B/Q(i)$ be ramified at $t<1+i>,$ $<2+3i>$ }. Let
be a maximal order in ,thenis a cocompact torsion-free Kleinian group of volume approximately 3.663862376708....
The quotientis the manifold $s705(1,2)$ in the SnapPea census of closed hyperbolic
3-manifolds. The first homology of this manifold is $Z/21Z$ .
$O$

$B$

$\Gamma_{\mathcal{O}}^{1}$$M=H^{3}/\Gamma_{\mathcal{O}}^{1}$

7

Geodesics in arithmetic hyperbolic 3-manifolds

Recall that if $M=H^{3}/\Gamma$ is a closed orientabk hyperbolic 3-manifold then the length spectrumof $M$ is the set of all lengths of closed $g\infty de8ics$ on $M$ counted with multiplicities. The length set$L(M)$ of $M$ is the set of lengths all closed $g\infty desioe$ counted without multiplicities. The mtional
length spectru $m\mathbb{Q}L(M)$ of $M$ is the set of all rational multiples of lengths of closed $g\infty desics$ of $M$ .
We shall denote the set of axes of hyperbolic elements in by.
This section is motivated by the following questions; in these, $M_{j}=H^{3}/\Gamma_{j}(j=1,2)$ we closedorientable hyperbolic 3-manifolds.
$\mathcal{L}(M)$$\Gamma$$A(\Gamma)$are$M_{1}$

and$M_{2}$are commensurable?

Question 7.2. If $L(M_{1})=L(M_{2})$ , are$M_{1}$

and$M_{2}$are commensumble?

Question 7.1. If
Question 7.3. If$\mathcal{L}(M_{1})=\mathcal{L}(M_{2})_{f}$$\mathbb{Q}L(M_{1})=\mathbb{Q}L(M_{2})$, are$M_{1}$

and$M_{2}$are commensumble?

Note that if
andare commensurable then, and that a positive answer
to Question 7.2 provides positive answers to Questions 7.1 and 7.3.
In addition, 8ince it is known (sae [16] pp. 415-417) that for closed hyperbolic manifolds, thespectrum of the Laplace.Beltrami operator action on $L^{2}(M)$ , counting multiplicities, determines$L(M)$ , a positive answer to Question 7.2 implies a positive answer to the next question. It 8hould be
noted that the currently known methods of producing $i8ospectral$ (same spectrum of the LaplaceBeltrami operator counted with multiplcities) closed hyperbolic 3-manifolds produce commensurable
ones (see [51] and [54]).
$M_{1}$

Question 7.4. If$M_{2}$$M_{1}$

and$\mathbb{Q}L(M_{1})=\mathbb{Q}L(M_{2})$$M_{2}$are isospectrd, are they commensumble?

“Dual” to these questions about lengths of $g\infty daeics$ i8 the foUowing question about axes determining commensurability.
Question 7.5. If $A(\Gamma_{1})=A(\Gamma_{2})$ , are$\Gamma_{1}$and$\Gamma_{2}$commensurable?

A8 above, it is euy to see that if andare commensurable, then.
Addressing these questions for general closed hyperbolic 3-manifolds seems very hard at present,
but arithmetic manifolds again provide an interesting class where these questions can be answered.
$\Gamma_{1}$$A(\Gamma_{1})=A(\Gamma_{2})$$\Gamma_{2}$Indeed, these questions can be uked for closed hyperbolic manifolds in arbitrary dimensisons, andconsiderable progress $hu$ been made recently in dimensioms
in the withmetic $cue$ by Pruad$\geq 4$and Rapinchuk [44]. In pwticular, they show that there exist arithmetic hyperbolic 5-manifolds
such that Question 7.3 hae a negative $an8wer$ .
In the remainder of this section we outline proofs of positive answers to all of the above questioIls
for arithmetic hyperbolic 2 and 3-manifolds. This 8urveys work in [8], [33] and [46].

53

7.1
We begin by sketching the proof of the following result in dimension 2. This is a mild extension of
the result in [46].
Theorem 7.6. Let $M_{1}=H^{2}/\Gamma_{1}$ be an onentable arithmetic hyperbolic lmanifold and assume
that $M_{2}=H^{2}/\Gamma_{2}$ is a dosed orientable hyperbolic 2-manifold with $L(\Gamma_{1})=L(\Gamma_{2})$ . Thenisarithmetic, andandare commensuroble.
$\Gamma_{2}$$\Gamma_{1}$$\Gamma_{2}$

Before commencing with the sketch of the proof, we discuss, what is perhaps the key featurethat aUows one to use withmetic methods. We motivate this with an example
Example: Let $\gamma\in PSL(2, Z)$ be a hyperbolic element of traoe$t$. Then the eigenvalues of$\gamma$are$\frac{(-t\pm\sqrt{t^{z}-4})}{beth^{2}e},andthaeebeingroo_{1}tsof.thepolynomialp(x)=x^{2}-tx+1.Let\lambda_{\gamma}--\frac{(-t+\sqrt{t-})}{s^{2}urface}eigenva1uesati\S\Psi ing|\lambda_{\gamma 1}>1Thelengthofthec1osedg\infty daeiconthemodu1ar$determined by the projection of the axis of is 21n.
Note thatis a unit, and it is an elementwy excercise to show that $p(x)$ is irreducible over Z.
Thus
determines a quadratic extension of Q.
Now $PSL(2, Z)$ coincides with the projection of the image of the elements of norm 1 in $M(2, Z)$ .
Thus we can find an element $u\in M(2, Q)$ such that $P(u)=\gamma$ , and we deduce from this thatembeds in $M(2, Q)uQ(u)$ .
Conversely, given a real quadratic extension of $Q$ that embeds in $M(2, Q)$ we can $\infty n8tuct$ ahyperbolic element $\gamma\in PSL(2, Z)$ for which
follows. Since is is a real quadratic
extension of $Q$ , the unit groupof $Lhu$ rank one. Choosing a fundamental unit , thenfor all integers $n\neq 0$ , and so $L=Q(u^{n})$ for all integers $n\neq 0$ . Now embeds in $M(2, Q)$ , so $Ri$
embeds in $M(2, Q)$ The key claim is that we can construct an order $O\subset M(2, Q)$ for which $u\in O$(sae [39] Chapter 12). It follows that since either or $u^{2}hu$ norm 1, we can wrange that oris an element of. By commensurability, we deduoe that there is an integer $m\neq 0$ such that$u^{m}\in SL(2, Z)$ and the image of this element in $PSL(2, Z)$ is the required hyperbolic element.
$H^{2}/PSL(2, Z)$$|\lambda_{\gamma}|$$\gamma$$\lambda_{\gamma}$$Q(\lambda_{\gamma})$$Q(\lambda_{\gamma})$

$L$

$L$

$L\cong Q(\lambda_{\gamma})u$$u^{n}\not\in Q$$u$$R_{L}^{*}$

$L$

$u$

$u^{2}$$u$$O^{1}$

That the quadratic extension is real is important, since otherwise, the unit group is finite, and wecannot $\infty n8truct$ a hyperbolic element as needed.
Extending this, the following facts are prov\’e in [39, Chapter12]. In the statement is either
an withmetic Fuchsian or Kleinian group.
$\Gamma$

Theorem 7.7. Suppose that(i) Letfield of$\Gamma$is derived fiom a quatemion algebm $B/k$ .

be a hyperbolic dement of
which embeds into $B$ .

$\gamma$$k$$\Gamma$

Utth eigenvalue.

$\lambda_{\gamma}$. The field$k(\lambda_{\gamma})$

is a quadratic extension(ii) Let be a quadratic extension of
If is Puchsian assume that is not a totdly imaginary
quadratic extension of .
Then embeds in $B/k$ if and only if
for some $hy\mu rbolic\gamma\in\Gamma$ . Ibis utll be true if
and only if no place of which splits in is mmified in $B$ .
$L$

$k$

$L$

$\Gamma$$k$$L$$L=k(\lambda_{\gamma})$

$k$

$L$

Sketch proof of Theorem 7.6: We begin by showing thatis arithmetic. Let the invariant
quaternion algebra of. Standwd considerations allow us to deduce that the set8 of traces
beof elements inandare the sa エロ e uP to sign, and that the sets {tr } we the same. Sinceis withmetic, tr
is an algebraic integer in the totally real field . Thus all elements ofhave algebraic integer trace and $k\Gamma_{2}=k$ . To complete the proof thatis arithmetic we show
thatis ramified at au infinite Placae except one, for then the characterization $th\infty rem$ of
Takeuchi [52] aPpliae. To see this we wgue as follows.
$\Gamma_{2}$$\Gamma_{1}$$\Gamma_{1}$$\Gamma_{1}$$B_{1}/k$$\gamma^{2}$$\Gamma_{2}$$\gamma^{2}$$k$$\Gamma_{2}$$\Gamma_{2}$$B_{2}=A\Gamma_{2}$

54

Let
for

$\alpha_{1}$$A\Gamma_{2}^{(2)}$and$\alpha_{2}$be a pair non-commuting hyperbolic elements incan be computed ae$( \frac{tr^{2}(\alpha_{1})-4_{I}tr[\alpha_{1},\alpha z[-2}{k})$$r_{2}^{(2)}$. Now a Hilbert Symbol(see [39] Chapter 3). Since$\Gamma_{1}$and$\Gamma_{2}$havethe same sets of traces (up to sign), we can flndsuch that $tr(\beta_{1})=\pm tr(\alpha_{1})$ and. It follows from [42] $Th\infty rem2.2$ thatsinoe.
Let : $karrow R$ be a non-identity embedding. Since
is ramified at it follows that $|\sigma(trx)|<2$
for all non-trivial elements $x\in B^{1}$ . Henoe we conclude that$\beta_{1},$$\beta_{2}\in\Gamma_{1}$$tr(\beta_{2})=\pm tr([\alpha_{1},\alpha_{2}])$$tr(\beta_{1}),tr(\beta_{2})\in k$$\beta_{1},\beta_{2}\in\Gamma_{1}\cap B_{1}^{1}$$B_{1}$$\sigma$$\sigma(tr^{2}(\beta_{1})-4)<0$$\sigma$and $\sigma(tr(\beta_{2})-2)<0$.

Henoe it foUows that$\sigma(tr^{2}(\alpha_{1})-4)<0$

and $\sigma(tr([\alpha_{1},\alpha_{2}])-2)<0$ ,from which it follows thatis ramified at . Now wae an wbitrary non-identity embedding,and so we have shown thatis ramified at $aU$ such embeddings a8 required.
$b$ establish commensurabihty, it suffices to show that(see [39] $Th\infty rem8.4.6$). To
this end, we make the following definition.
$B_{2}$$\sigma$$\sigma$$B_{2}$$B_{1}\simeq B_{2}$Deflnition: For $j=1,2$ let$\mathcal{N}_{j}=${ $L/k;[L:k]=2,$$L$

emebds in$B_{j}$and is not a totally imaginary quadratic extension of

$k$

}.

and
are isomorphic if and only if the set of quadratic extensions that embed in one
is precisely the set that embeds in the other. The key claim that needs to be shown is:
Clewly$B_{1}$$B_{2}$$B_{1}$

and$B_{2}$are isomorphic if and only $N_{1}=N_{2}$This is established in [46]. Given this, the proof ls complet\’e as foUows. Suppose $L\in N_{1}$ . $Th\infty rem$
7.7 and commen8urabi1ity shows thatfor some hyperbolicwith eigenvalue .
Since $L(M_{1})=L(M_{2})$ , it follow8 that there exists an element
for which. Henceembeds inas required.
$L=k(\lambda_{\gamma})$$\gamma\in F_{1}$$\gamma’\in\Gamma_{2}$$L=k(\lambda_{\gamma})=k(\lambda_{\gamma’})$$B_{2}$$\lambda_{\gamma}$$\lambda_{\gamma’}=\lambda_{\gamma}$$\square$

In dimension 3, a similar result is established in [46] for the complex length spectrum. However,to deal with lengths only requires considerably more work. The foUowing is proved in [8], and in
pwticular gives an affirmative answer to Questions 7.1, 7.27.3 and 7.4 whenand
we botharithmetic.
$M_{1}$$M_{2}$

Theorem 7.8. If $M$ is an arithmetic hyperbolic 3-manifold, then the rationd length spectrum andthe commensumbility class of $M$ determine one another.
The proof of $Th\infty rem7.8$ is modeUed on some of the ideae $\infty ntained$ in the proof of $Th\infty rem$
7.6, but more involved, since one cannot deal with traces directly. Briefly, the first step in proving
that
determines the commensurabillty claes is to show that the invarlant trace-field isdetermined by $QL(M)$ We then determine the commensurability class of $M$ fromfollowingideae similar to those used in the proof of $Th\infty rem7.6$. The wguments for the determination of
splits naturaUy into two cases; when $[k : k\cap R]>2$ and $[k : k\cap R]=2$ . Both cases requirea detailed understanding of the Galois $th\infty ry$ of fields with one complex
and their quadratic
extension8. We refer the reader to [8] for details.
$\mathbb{Q}L(M)$.

$\mathbb{Q}L(M)$$k\Gamma$$pla\infty$

55

7.2
If we consider only the “location” of the axos of hyperbolic elements in arithmetic Fuchsian and
Kleinian groups, then again one can get extra information that allows one to prove (see [33]).
Theorem 7.9. If
and
has a positive answer.
$\Gamma_{1}$$\Gamma_{2}$are arithmetic $fb$ chsian (resp. Kleinian) groups then Question 7.5

For convenience, we shall call groups
and.
isoaxial if
As with the proof of $Th\infty rem7.6$ the key issue is to show that the isoaxial condition forces
the same invariant quaternion algebra. We sketch the proof for Kleinian groups that requires thefollowing definitions. The obvious changes can be made for Fuchsian groups.
Let be a Kleinian group and define$\Gamma_{1}$$\Gamma_{2}$$A(\Gamma_{1})=A(\Gamma_{2})$$\Gamma$.

$\Sigma(\Gamma)=\{\gamma\in PSL(2, C)|\gamma(A(\Gamma))=A(\Gamma)\}$It is easy to check thatis a subgroup of $PSL(2, C)$ , and that Comm
. The key
point now is to show that when is withmetic, then Comm
. Given this, the proof iscompleted ae follows.
If andare isoaxial Kleinian groups, then for any, and therefore. Henoe. In addition, if is withmetic, then the equality Comm
implies that $\Gamma_{2}<Comm(\Gamma_{1})$ . It is now standard that, sinoe
is also withmetic,andare$\Sigma(\Gamma)$$(\Gamma)<\Sigma(\Gamma)$$(\Gamma)=\Sigma(\Gamma)$$\Gamma$$\Gamma_{1}$$\Gamma_{2}$$\gamma\in\Gamma_{2},$$\gamma\in\Sigma(\Gamma_{1})$$\Gamma_{2}<\Sigma(\Gamma_{1})$$A(\Gamma_{1})=\mathcal{A}(\gamma\Gamma_{1}\gamma^{-1})$$(\Gamma_{1})=\Sigma(\Gamma_{1})$$\Gamma_{1}$$\Gamma_{2}$commensurable.

$\Gamma_{1}$$\Gamma_{2}$$\square$Remark: In [35] we construct examples of Fuchsian groups of finite co-wea that have the same setof parabolic fixed polnts $PSL(2, Z)$ but we non-arithmetic. The question of whether these Fuchsiangroups share the same set of axes $u$ hyperbolic element8 of $PSL(2, Z)$ remains open, although wesuspect that this is not the caee.

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$p$

Department of Mathematics,
University of Texae
Austin, TX 78712, U.S.A.
Email: weid@math.utexu.\’eu