Table Of ContentMACFARLANE HYPERBOLIC 3-MANIFOLDS
JOSEPH A. QUINN
7
1 Abstract. Weidentifyandstudyaclassofhyperbolic3-manifolds(whichwecallMacfar-
0 lane manifolds) whose quaternion algebras admit a geometric interpretation analogous to
2 Hamilton’s classical model for Euclidean rotations. We characterize these manifolds arith-
n metically, and show that infinitely many commensurability classes of them arise in diverse
a topologicalandarithmeticsettings. Wethenusethisperspectivetointroduceanewmethod
J forcomputingtheirDirichletdomains. Wealsogivesimilarresultsforaclassofhyperbolic
4 surfaces and explore their occurrence as subsurfaces of Macfarlane manifolds.
2
] 1. Introduction
T
G Quaternion algebras over complex number fields arise as arithmetic invariants of complete
. orientable finite-volume hyperbolic 3-manifolds [5, 15]. Quaternion algebras over totally
h
real number fields are similarly associated to immersed totally-geodesic hyperbolic subsur-
t
a
faces of these manifolds [15, 27]. The arithmetic properties of the quaternion algebras can
m
be analyzed to yield geometric and topological information about the manifolds and their
[
commensurability classes [16, 18].
1 In this paper we introduce an alternative geometric interpretation of these algebras, re-
v calling that they are a generalization of the classical quaternions H of Hamilton. In [20],
2
1 the author elaborated on a classical idea of Macfarlane [14] to show how an involution on
7 the complex quaternion algebra can be used to realize the action of Isom`pH3q multiplica-
6 tively, similarly to the classical use of the standard involution on H to realize the action of
0
. Isom`pS2q. Here we generalize this to a class of quaternion algebras over complex number
1
fields and characterize them by an arithmetic condition. We define Macfarlane manifolds as
0
7 those having these algebras as their invariants.
1 We establish the existence of arithmetic and non-arithmetic Macfarlane manifolds and in
:
v each of these classes, infinitely many non-commensurable compact and non-compact exam-
i ples. Wethengeneralizethecomplexquaternionictoolsfrom[20]todevelopanewalgorithm
X
forcomputingDirichletdomainsofMacfarlanemanifoldsandtheirimmersedtotally-geodesic
r
a hyperbolic subsurfaces.
Main Results. Let X be a complete orientable finite-volume hyperbolic 3-manifold. Let
K and B be the trace field and quaternion algebra of X, respectively.
Definition 1.1. X is Macfarlane if and only if
(1) K is an imaginary quadratic extension of a real field, and
(2) complex conjugation acts freely on the ramification set of B.
The main idea (made precise by Theorem 3.3 and Corollary 4.2) can be stated informally
as follows.
Date: January 25, 2017.
1
2 JOSEPHA.QUINN
Theorem 1.2. X is Macfarlane if and only if there exists an involution : on B which,
with the quaternion norm, naturally gives rise to a 3-dimensional hyperboloid M1 Ă B over
`
K XR. Moreover, : is unique and the action of π pXq by orientation-preserving isometries
1
of H3 can be written quaternionically as
π pXq ü M1, pγ,pq ÞÑ γpγ:.
1 `
By comparison, via Hamilton’s classical result one can use the standard involution ˚ on
H to realize Isom`pS2q quaternionically [20] as
PH1 ü H1, pγ,pq ÞÑ γpγ˚.
0
In §4, we describe an adaptation of the main result for hyperbolic surfaces and show how,
in certain instances, an immersion of a surface in a 3-manifold is sufficient for the 3-manifold
to be Macfarlane. We then study other topological and arithmetic conditions under which
Macfarlane manifolds arise, culminating in the following theorem.
Theorem 1.3. There exist infinitely many non-commensurable Macfarlane manifolds in
each of the following categories:
(1) arithmetic and non-compact,
(2) arithmetic and compact,
(3) non-arithmetic and non-compact,
(4) non-arithmetic and compact.
Lastly, in §5 we use the quaternion model to provide a new algorithm for computing
Dirichlet domains and illustrate the technique with some basic examples.
2. Preliminaries
See [4, 21] for preliminary information on Kleinian and Fuchsian groups and hyperbolic
topology. See[20]forpreliminaryinformationonalgebraswithinvolution,thestandardMac-
farlane space and some additional historical context. See [30] for a comprehensive treatment
of quaternion algebras.
2.1. Quaternion algebras. Let K be a field with charpKq ‰ 2. In this article K will
usually be one of: R,C, a p-adic field, or a concrete number field i.e. rK : Qs ă 8 with a
fixed embedding K Ă C.
Definit´ion¯2.1. Leta,b P Kˆ, calledthestructure constants ofthealgebra. Thequaternion
algebra a,b istheassociativeK-algebra(withunity)K‘Ki‘Kj‘Kij,withmultiplication
K
rules i2 “ a, j2 “ b and ij “ ´ji.
(1) The quaternion conjugate of q is q˚ :“ w´xi´yj ´zk.
(2) The (reduced) norm of q is npqq :“ qq˚ “ w2´ax2´by2`abz2.
(3) The (reduced) trace of q is trpqq :“ q`q˚ “ 2w.
(4) q is a pure quaternion when trpqq “ 0.
´ ¯
We indicate conditions on the trace via subscript, for instance given a subset E Ă a,b ,
(cid:32) ( (cid:32) ( K
we write E0 “ q P E | trpqq “ 0 and E` “(cid:32) q P E | trpqq ą(0 . We indicate conditions on
the norm via superscript, for instance E1 “ q P E | npqq “ 1 .
´ ¯
Proposition2.2. [30]IfK Ă C, thenDafaithfulmatrixrepresentationof a,b intoM pCq.
K 2
Moreover, under any such representation, n and tr correspond to the matrix determinant and
trace, respectively.
3
´ ¯
We will be interested in the K-algebra isomorphism class of a,b , which is not uniquely
K
determined by a and b.
Theorem 2.3. [30]
´ ¯ ´ ¯
(1) Either a,b – M pKq, or a,b is a division algebra.
´ ¯ K 2 K
(2) a,b – M pKq ðñ D px,yq P K2 such that ax2`by2 “ 1.
K 2 ´ ¯ ´ ¯ ´ ¯
(3) For all x P Kˆ, a,b – b,a – ax2,b .
K K K
´ ¯
It follows that for any K, there is the quaternion K-algebra 1,1 – M pKq. Moreover, if
´ ¯ K 2
a,b is not a division algebra, then its isomorphism class is unique and can be represented
K´ ¯
by 1,1 . Thus we now focus on quaternion division algebras.
K
Example 2.4. See [30] for proofs of (3) and (4) below.
´ ¯
(1) Over R, the only quaternion division algebra up to isomorphism is H :“ ´1,´1 ,
R
Hamilton’s quaternions.
(2) There are no quaternion division algebras over C.
(3) Over a p-adic field, there is a unique quaternion division algebra up to isomorphism.
(4) Over a number field, there are infinitely many non-isomorphic quaternion division
algebras.
This raises the question of how to tell, when K is a number field, whether or not two
quaternion K-algebras are isomorphic. This can be done by investigating the local algebras
with respect to the places of K, in the following sen´se.¯
Let K be a (concrete) number field and let B “ a,b . For a place v of K, let K be the
K v
completion of K with respect to v. To each v, we associate an embedding σ : K ãÑ K as
v
described in the following paragraph, and then define the localization of B with respect to v
as B :“ Bb K .
v σ v
If v is infinite (i.e. Archimedean), then it corresponds (up to complex conjugation) to
an element of the Galois group of K over Q, under which the completion of the image of
K is either´R or C,¯and we define σ as the corresponding embedding. So if σpKq Ă R
then B “ σpaq,σpbq , which is isomorphic to either H or M pRq. If σpKq Ę R then B “
´ v ¯ R 2 v
σpaq,σpbq , which is always isomorphic to M pCq. If v is finite (i.e. non-Archimedean), then
C 2
it corresponds to a prime ideal P Ÿ ZK, where ZK is the ring of integers of K. In th´is ca¯se
we define σ as the identity embedding into the corresponding p-adic field, thus B :“ a,b .
v KP
Definition 2.5.
(1) B is ramified if it is a division algebra, and is split if B – M pKq.
2
(2) B is ramified at v (respectively, split at v) if B is ramified (respectively, split).
v
(3) RampBq is the set of real embeddings and prime ideals that correspond to the places
where B is ramified.
The set RampBq provides the desired classification of isomorphism classes of quaternion
algebras over number fields.
4 JOSEPHA.QUINN
Theorem 2.6. [30]
(1) B is split if and only if RampBq “ Ø.
(2) RampBq uniquely determines the isomorphism class of B.
(3) RampBq is a finite set of even cardinality, and every such set of places of K occurs
as RampBq for some B.
2.2. The arithmetic of hyperbolic 3-manifolds. Let X be a complete orientable finite-
volume hyperbolic 3-manifold. T(cid:32)hen π1pXq – Γ ă( PSL2pCq for some discrete group Γ (i.e.
Γ is a Kleinian group). Let Γp :“ ˘γ | t˘γu P Γ ă SL pCq.
2
Definition 2.7. `(cid:32) (˘
(1) The trace field of Γ is KΓ :“ Q trpγq | γ P Γp .
(cid:32)ř (
(2) The quaternion algebra of Γ is BΓ :“ n t γ | t P KΓ,γ P Γp,n P N .
(cid:96)“1 (cid:96) (cid:96) (cid:96) (cid:96)
Remark 2.8. In the literature these are usually denoted by k Γ and A Γ, but we write
0 0
them differently to avoid confusion with the notation for pure quaternions.
KΓ is a number field and BΓ is a quaternion algebra over KΓ [15]. By Mostow-Prasad
rigidity, these are manifold invariants in the sense that if Γ and Γ1 are two faithful represen-
tations of π pXq, then KΓ “ KΓ1 and BΓ – BΓ1 via a KΓ-algebra isomorphism (though
1
the converse does not hold). So we may also refer to them as the trace field and quaternion
algebra of X up to homeomorphism.
Definition 2.9. Let Γp2q :“ xγ2 | γ P Γy.
(1) The invariant trace field of Γ is kΓ :“ KΓp2q.
(2) The invariant quaternion algebra of Γ is AΓ :“ BΓp2q.
These likewise are invariants of X, but have the stronger property of being commensura-
bility invariants. That is, if Γ is commensurable up to conjugation to some Kleinian group
Γ1, then kΓ “ kΓ1 and AΓ – AΓ1 (though the converse does not hold) [18].
We call X arithmetic if Γ is an arithmetic group in the sense of [6], but this admits the
following alternative characterization [15].
Definition 2.10.
(1) Γ (or X) is derived from a quaternion algebra if there exists a quaternion algebra B
over a field K with exactly one complex place σ, such that B is ramified at every real
place of K, and D an order O Ă B such that Γ is a finite-index subgroup of PσpOq1.
(2) Γ (or X) is arithmetic if it is commensurable up to conjugation to one that is derived
from a quaternion algebra.
If Γ is derived from a quaternion algebra B over a field K, then K “ KΓ “ kΓ and
B “ BΓ “ AΓ. If Γ is arithmetic, then Γp2q is derived from a quaternion algebra, so then
AΓp2q “ BΓp2q. In general, Γp2q is a finite-index subgroup of Γ. [18]
While kΓ and AΓ are generally more suitable to the application of arithmetic, we will
p
work instead with BΓ so that we may take advantage of the natural embedding Γ ãÑ BΓ.
(To simplify notation, and where it will not cause confusion, we will often refer to an element
p
t˘γu P Γ by a representative γ P Γ.) Often, AΓ and BΓ coincide (though not always [22]).
Proposition 2.11.
(1) kΓ “ KΓ if and only if AΓ – BΓ.
(2) If H3{Γ is a knot or link complement, then AΓ – BΓ.
5
Proof. We prove (1), and see [16] for (2). The reverse implication is immediate. For the
forward implication, note that AΓ Ă BΓ and both are 4-dimensional vector spaces, so if they
are over the same field then they must be the same. (cid:3)
We now collect some important properties of these invariants.
Theorem 2.12. [16]
´ ¯ ´ ¯
(1) If X is non-compact, then BΓ – 1,1 and AΓ – 1,1
KΓ kΓ
?
(2) If X is non-compact and arithmetic, then Dd P N such that kΓ “ Qp ´dq.
(3) If X is compact and arithmetic, then AΓ is a division algebra.
3. Macfarlane Quaternion Algebras and Isom`pH3q
Our goal in this section is to show that the arithmetic characterization of Macfarlane
manifolds given by Definition 1.1 admits the geometric interpretation given by Theorem 1.2.
If B is the quaternion algebra of a Macfarlane manifold, we will call B Macfarlane as well.
To define this property more generally, let B be a quaternion algebra over a field K Ă C (not
necessarily a number field).
Definition 3.1. B is Macfarlane if
?
(1) DF Ă R and Dd P F` such that K “ Fp ´dq, and
(2) complex conjugation acts freely on RampBq, i.e. is closed and has no fixed points.
Examp´le 3¯.2. ´ ¯
?
(1) 1,1 is Macfarlane because C “ Rp ´1q and Ram 1,1 “ Ø.
C C ´ ¯
(2) The figure-8 knot complement and it’s quaternion algebra ?1,1 are Macfarlane.
? (cid:32) Qp ´?3q ? (
(3) The quaternion algebra B over Qp ´5q with RampBq “ p3,1` ´5q,p3,1´ ´5q
is Macfarlane, and so is any manifold derived from it.
We now give a result which is similar to and implies Theorem 1.2, but in terms of B and
with more detail. First notice that by Proposition 2.2, PB1 is isomorphic to a subgroup
of PSL pCq. This implies an injection PB1 ãÑ Isom`pH3q, but for Macfarlane quaternion
2
algebras we can make this explicit.
Theorem 3.3. B is Macfarlane if and only if it admits an involution : such that SympB,:q
(which we denote by M), equipped with the restriction of the quaternion norm, is a quadratic
space of signature p1,3q over SympK,:q. (cid:32) (
Moreover, : is unique and, letting M1 “ p P M | trppq ą 0,nppq “ 1 , a faithful action
`
of PB1 upon H3 by orientation-preserving isometries is defined by the group action
µ : PB1 ü M1, pγ,pq ÞÑ γpγ:.
B `
Definition 3.4. M as in Theorem 3.3 is called a Macfarlane space.
Proof of Theorem 3.3. First we show that the existence of an involution as in the The-
orem is equivalent to a condition on the field and structure constants of the algebra, up to
isomorphism.
Lemma 3.5´. B adm¯its an involution with the properties described in the Theorem if and
only if B – ?a,b for some F Ă R and a,b,d P F`.
Fp ´dq
6 JOSEPHA.QUINN
´ ¯
The reverse direction of this, in the case where B “ ?a,b , is Theorem 7.2 of [20].
´ ¯ Fp ´dq
This generalizes to B – ?a,b because an isomorphism between quaternion algebras is
Fp ´dq
also a quadratic space isometry with respect to the quaternion norms [30], thus it transfers
the multiplicative structure, the involution and the Macfarlane space.
So it suffices to prove the forward direction, and we do this via a series of claims. Let
B be a quaternion algebra over a field K and suppose B admits an involution : with the
properties described in Theorem 3.3.
?
Claim 3.6. K is of the form Fp ´dq where F “ SympK,:q Ă R and d P F`, and :| is
K
complex conjugation.
Proof. If K were real, then n would be a quadratic form of signature p2,2q, making it
impossible for B to contain a subspace of signature p1,3q, thus K Ę R. On the other hand,
for a space to have nontrivial signature over SympK,:q, we must have SympK,:q Ă R. This
means : is an involution of the second kind which implies rK : SympK,:qs “ 2 [13, 20], i.e.
?
K “ Fp ´dq as in the Claim.
We now show that :| is complex conjugation. Since ´d P F “ SympK,:q, we have
K
? ?
p ´d:q2 “ p ´d2q: “ p´dq: “ ´d,
? ? ? ? ?
thus ´d: “ ˘ ´d. Since ´d R SympK,:q, this leaves ´d: “ ´ ´d. (cid:3)
Write M “ SympB,:q. We are going to use the signature of n| to prove that B has real
M
structure parameters up to isomorphism, but a priori we do not know what M is. So we
will first need to establish that M includes enough linearly independent elements of B, in
the following sense.
Claim 3.7. Span pMq “ B.
K
Proof. We know that F Ă M and Span pFq “ K, so it suffices to prove Span pM q “ B .
K K 0 0
Let E “ ts1,s2,s3u be a basis for M0 over F and assume by wayřof contradiction that
E is not linearly independent over K. Then D k P K such that 3 k s “ 0. Since
? ?(cid:96) (cid:96)“1 (cid:96) (cid:96)
K “ Fp ´dq, we have that each k “ f `f ´d for some f ,f P F. Substituting
ř (cid:96) (cid:96),1 (cid:96),2 (cid:96),1 (cid:96),2
these into 3 k s “ 0 and rearranging terms, we get
(cid:96)“1 (cid:96) (cid:96)
?
f s `f s `f s “ ´ ´dpf s `f s `f s q.
1,1 1 2,1 2 3,1 3 1,2 1 2,2 2 3,2 3
Butf s `f s `f s andf s `f s `f s bothlieinM,soarefixedby:,meanwhile
1,1 1 2,1 2 3,1 3 1,2 1 2,2 2 3,2 3
? ?
:
by the previous claim, ´d “ ´ ´d. So applying : to both sides of the equation gives
?
f s `f s `f s “ ´dpf s `f s `f s q.
1,1 1 2,1 2 3,1 3 1,2 1 2,2 2 3,2 3
Adding the last two displayed equations then gives that f s `f s `f s “ 0. Since
1,2 1 2,2 2 3,2 3
f ,f ,f P F, this contradicts that E is a basis for M over F.
1,2 2,2 3,2 0
We conclude that E is linearly independent over K, giving that
` ˘ ` ˘
dim Span pEq “ dim Span pM q “ 3,
K K K K 0
which forces Span pM q “ B , as desired. (cid:3)
K 0 0
´ ¯
Claim 3.8. B – ?a,b for some a,b P F`.
Fp ´dq
7
Proof. The norm n| is a real-valued quadratic form of signature p1,3q, so there exists
M
an orthogonal basis D for M so that the Graham matrix for n| with respect to D is a
M
diagonal matrix GD , with diagonal of the form pf ,´f ,´f ,´f q for some f P F`. Since
n|M 1 2 3 4 (cid:96)
Span pMq “ B, this same D is also an orthogonal basis for B over K.
K
Let C be the standard basis t1,i,j,iju for B. Then C is another orthogonal basis for B
over K and, in particular, the Graham matrix GC for n with respect to C is the diagonal
n
matrix with diagonal p1,´a,´b,abq.
Now while GD and GC are not congruent over F, they are congruent over K because D
n|M n
and C are both bases for B, i.e. D δ P GL pKq such that
4
(3.9) δGDδJ “ GC.
n n
But since GD and GC are diagonal and nonzero on their diagonals, δ must also be diagonal
n n
and nonzero on its diagonal, i.e. D x P Kˆ such that the diagonal of δ is px ,x ,x ,x q.
(cid:96) 1 2 3 4
Plugging in to (3.9) and solving for the f gives
(cid:96)
1 a b ´ab
f “ f “ f “ f “ .
1 x2 2 x2 3 x2 4 x2
´ ¯ 1 2 3 4
Now let B1 “ f?2,f3 and recall that f2,f3 P F`. Then B has the desired form because,
Fp ´dq
by Theorem 2.3,
´ ¯
f x2,f x2
B1 – 2 ?2 3 3 “ B.
Fp ´dq
(cid:3)
This completes the proof of Lemma 3.5.
Nowtocomplete´thep¯roofofTheorem3.3,weshowthattheconditionontheisomorphism
class of the symbol a,b from Lemma 3.5 is equivalent to the arithmetic characterization of
K
Macfarlane quaternion algebras given by Definition 3.1.
?
Lemma 3.10. Let B be a quaternion algebra over K “ Fp ´dq where F Ă R and ´d P F¯`.
Complex conjugation acts freely on RampBq if and only if Da,b P F` such that B – a,b .
K
´ ¯ ´ ¯
Proof. With a,b,F and K as in the statement, notice that a,b “ a,b b K. Also if a
K F F
(or b) is negative, then by Theorem 2.3, we can replace it by ´ad (or ´bd) without changing
the isomorphism class. So it suffices to prove that the condition on RampBq is equivalent to
the existence of a quaternion algebra A over F such that B – Ab K.
F
Since K is an imaginary quadratic extension of a real field, it has no real places, making
RampBq a set of prime ideals of Z . Denote an arbitary prime ideal of Z by P. So
K K
for complex conjugation to act freely on RampBq is to say that @P P RampBq, we have
P ‰ P P RampBq.
ForeveryprimeidealP Ÿ Z , thereexistsaprimeidealp Ÿ Z suchthatP|pwherepZ
K F K
is one of P, P2, or PP where P ‰ P (i.e. p is inert, ramified or split in the extension K : F,
respectively). In the first two cases P “ P, and the third possibility is the only way that
these prime ideals occur in complex conjugate pairs. So to say that complex conjugation
acts freely on RampBq is equivalent to saying that there exist prime ideals p ,...,p Ÿ Z
1 n F
such that
(cid:32) (
(3.11) RampBq “ P ,P | p Z “ P P , P ‰ P, (cid:96) “ 1,...,n .
(cid:96) (cid:96) (cid:96) K (cid:96) (cid:96)
8 JOSEPHA.QUINN
Our task is thus to prove that RampBq has this form if and only if D a quaternion algebra A
over F such that B – Ab K.
F
For the reverse direction, suppose that B – A b K as above. As already noted, any
F
infinite place of A becomes complex in B, so does not contribute to RampBq. So let p Ÿ Z
F
be a prime ideal and we investigate its contribution to RampBq. If p R RampAq, i.e. A is
p
split, thenB issplitattheprimeslyingabovep, meaningthatthosedonotoccurinRampBq.
So we can assume p P RampAq.
If pZ has the form P or P2, then K – F pxq{px2 `dq, a quadratic extension of F .
K P p p
Since F is a p-adic field, A contains all of its quadratic extensions [30]. Then K ãÑ A ,
p p P p
which implies that A b K is split, but this equals B . Thus P R RampBq.
p Fp P P
If pZ has the form PP, then K “ F , and then A – B (and likewise for P). So the
K P p p P
primes above p occur in RampBq, giving it the desired form.
For the forward implication, suppose Dp ,...,p Ÿ Z such that Equation (3.11) holds.
1 n F
FromTheorem2.3weknowthatforanyfiniteevensetofplacesofF,thereexistsaquaternion
algebra over F having that as its ramification set. If n is even, let A be the quaternion
algebra over F with RampAq “ tp ,...,p u. If n is odd, choose any one of the (infinitely
1 n
many) prime ideals of p Ÿ Z that does not split in the extension to K, and let A satisfy
F
RampAq “ tp,p ,...,p u. The above argument then gives that RampAb Kq “ RampBq,
1 n F
thus B – Ab K. (cid:3)
F
This completes the proof of Theorem 3.3. The following consequence has computational
advantages which will be exploited in §5.
´ ¯
Corollary 3.12. If B is Macfarlane, then there is an isomorphism B – ?a,b where
Fp ´dq
a,b,d P F`. In this case the Macfarlane space is
?
M “ F ‘Fi‘Fj ‘ ´dFij
?
and for q “ w`xi`yj `zij P B with w,x,y,z P Fp ´dq, the involution : is given by
(3.13) q: “ w`xi`yj ´zij.
A final remark on Theorem 3.3 is that even though we are using M1 as a hyperboloid
`
model for the group action of interest, it is technically not a model for H3 unless F “ R. We
will study the case where F is a concrete number field embedded in R. If a complete model
for H3 is desired, one is given by pMb Rq1 but we will generally not need this.
F `
4. Macfarlane Manifolds
In this section we explore the various conditions in which Macfarlane manifolds arise.
First we clarify why Theorem 1.2 about Macfarlane manifolds follows from Theorem 3.3
about Macfarlane quaternion algebras. Then we look at an adaptation of our results to
hyperbolic surfaces, and see how immersed subsurfaces sometimes give rise to Macfarlane
3-manifolds. The remainder of the section culminates in a proof of Theorem 1.3.
Let X denote a complete orientable finite-volume hyperbolic 3-manifold with Kleinian
group Γ – π pXq. Let K “ KΓ and B “ BΓ.
1
Definition 4.1. When X is Macfarlane and M Ă B is its Macfarlane space as in Theorem
3.3, define I :“ M1 and call this a quaternion hyperboloid model for Γ (or X).
Γ `
It is immediate that I , up to quadratic space isometry over KΓ, is a manifold invariant.
Γ
9
By the definition of BΓ “ B, there is no confusion in speaking of Γ quaternionically, as
lying in PB1 rather than in PSL pCq. In this way, Γ (up to choice of representatives in Γp)
2
and I are both subsets of B, making sense of the following, which gives Theorem 1.2.
Γ
Corollary 4.2. If X is Macfarlane, then the action of Γ by orientation-preserving isometries
of H3 is faithfully represented by
µ : Γ ü I , pγ,pq ÞÑ γpγ:.
Γ Γ
4.1. Hyperbolic Surfa´ces¯and Subsurfaces. Theauthorshowedin[20]h´owt¯herepresen-
tation of Isom`pH3q in 1,1 restricts to a representation of Isom`pH2q in 1,1 . Similarly,
C R
we seek an analogue of Theorem 1.2 for hyperbolic subsurfaces of 3-manifolds. This is pos-
sible to some extent but we must proceed with care because there are important differences
between the 3-dimensional and 2-dimensional settings.
Let S denote a complete orientable finite-volume hyperbolic surface. The group π pSq
1
admits discrete faithful representations into PSL pRq but in the absence of Mostow-Prasad
2
rigidity, an isomorphism class of such representations only gives S up to isometry, not home-
omorphism. Furthermore, a Fuchsian group can contain transcendental traces, meaning its
trace field need not be a number field. We will soon see a way of getting around this but for
now let us think of S up to isometry.
Let ∆ ă PSL pRq be a Fuchsian group. The (invariant) trace field of ∆ and (invariant)
2
quaternion algebra of ∆ are defined in the same way as in Definitions 2.7 and 2.9, and we
denote them similarly by (k∆) K∆ and (A∆) B∆, respectively. These have properties
similar to what we saw in the Kleinian setting. For instance, B∆ and A∆ are quaternion
algebras over K∆ and k∆ respectively [25], and A∆ is a commensurability invariant [28].
The results from §6 of [20] along with the proof of Lemma 3.5 then give the corollary
below, after the following observations. The field K∆ is real, and so now the involution : is
of the first kind. That is, SympK∆,:q “ K∆ and SympB∆,:q is comprised of K∆ and the
unique 2-dimensional negative-definite subspace with respect to the norm on B∆.
´ ¯
Corollary 4.3. If B∆ – a,b for some a,b ą 0, then it admits an involution : such that
K∆
SympB∆,:q (which we denote by L), equipped with the restriction of the quaternion norm,
is a quadratic space of signature p1,2q over(cid:32)K∆. (
Moreover, : is unique and, letting L1 “ p P L | trppq ą 0,nppq “ 1 , a faithful action of
`
∆ upon H2 by orientation-preserving isometries is defined by the group action
µ : ∆ ü L1, pγ,pq ÞÑ γpγ:.
∆ `
Definition 4.4. We call the space L Ă B∆ as above a restricted Macfarlane space, and we
call the space I :“ L1 a quaternion hyperboloid model for ∆.
∆ `
We next give a way of realizing Macfarlane 3-manifolds using Fuchsian groups. Despite
the absence of rigidity, in the homeomorphism class of S there always exists a representation
∆ ă PSL pRq of π pSq such that K∆ is a number field [26]. When S is an immersed
2 1
closed totally-geodesic subsurface of a hyperbolic 3-manifold X, there exists an injection
π pSq ãÑ π pXq, which implies such a ∆ [16]. In this context, we think of S as the isometry
1 1
class of surfaces induced by the homeomorphisms of X.
?
Proposition 4.5. If K “ Fp ´dq for some F Ă R and d P F`, and X contains an
immersed closed totally-geodesic surface, then X is Macfarlane.
10 JOSEPHA.QUINN
Proof. First let S Ă X be a surface as in (1). Then π pSq has a Fuchsian representation
1
∆ ă PSL pRqandπ pXqhasaKleinianrepresentationΓ ă PSL pCqsuchthat∆ ă Γ. Then
2 1 2
K∆ Ă K XR “ F. Therefore B´∆ Ă¯BΓ is a quaternion subalgeb´ra over¯a subfield of F.
Hence D a,b P F so tha´t B∆ “¯ Ka,∆b . But then B∆bK∆ K “ Fp?a,´bdq Ă BΓ. So by
Proposition 2.11, BΓ “ ?a,b , and then by Lemma 3.10, X is Macfarlane. (cid:3)
Fp ´dq
Remark 4.6. When an immersion occurs as above, the action µ as given in Theorem 1.2
Γ
restricts to the action µ as given in Corollary 4.3. An example of this will be studied in §5.
∆
4.2. Arithmetic Macfarlane Manifolds. For X to be arithmetic and Macfarlane, its
invariant trace field kΓ must be quadratic. To see this, recall that the trace field of a
?
Macfarlane manifold is Fp ´dq for some F Ă R and d P K`, and that the invariant trace
?
field is a complex subfield of this, forcing it to be of the form F1p ´d1q with F1 Ă F and
d1 P F1`. By Definition 2.10, the invariant trace field of an arithmetic Kleinian group can
have only one complex place, which forces F1 “ Q.
Quadratic fields do not have any real embeddings, therefore an order of a quaternion
algebra over one of these fields will always give rise to a Kleinian group derived from a
quaternion algebra. We use this to construct examples of arithmetic Macfarlane manifolds
and prove parts (1) and (2) of Theorem 1.3.
4.2.1. Non-compact Arithmetic Macfarlane Manifolds. Every arithmetic non-compact Γ is
commensurable to a Bianchi group [16], which is a group of the form PSL pZ ? q where
´ ¯2 Qp ´dq
d P N. Such a group is derived from the quaternion algebra ?1,1 . It follows that
Qp ´´dq ¯
whenever X is non-compact and arithmetic, AΓ will be of the form ?1,1 , so that if
Qp ´dq
KΓ “ kΓ, then X will be Macfarlane. (While this is often the case, it is not always [22]. For
?
exampl´ebthe man¯ifold m009 in the cusped census has invariant trace field Qp ´7q but trace
?
field Q 5´ ´7 .)
2
Lemma 4.7. There are infinitely many commensurability classes of non-compact arithmetic
Macfarlane manifolds.
Proof. For each square-free d P N, take a torsion-free subgroup Γ ă PSL pZ ? q (whose
2?Qp ´dq
existence is guaranteed by Selberg’s Lemma). I´f KΓ is n¯ot of the form Qp´´dq, th¯en form
Γp2q. Since Γ is arithmetic, Γp2q is derived from ?1,1 , so that BΓp2q “ ?1,1 , which
Qp ´dq Qp ´dq
isMacfarlanebyLemma3.5. SinceAΓp2q “ BΓp2q eachchoiceofdgivesagroupinadifferent
commensurability class. (cid:3)
Example 4.8. Arithmetic link complements are Macfarlane, by Proposition 2.11.
?
(1) The figure-8 knot complement is Macfarlane, with trace field Qp ´3q.
?
(2) The Whitehead link is Macfarlane, with trace field Qp ´1q.
?
(3) The six-component chain link is Macfarlane, with trace field Qp ´15q.
4.2.2. Compact Arithmetic Macfarlane Manifolds. To construct these we again work over
the quadratic fields. While there is only one split quaternion algebra over each field, there
are infinitely many ramified ones. We use this to get something stronger than just a direct
analog of Lemma 4.7.
Lemma 4.9. For every square-free d P N, there are infinitely many commensurability classes
?
of compact arithmetic Macfarlane manifolds having the trace field Qp ´dq.
