On the number of ends of rank one locally symmetric spaces

0.22 MB·English

by  Matthew Stover

On the number of ends of rank one locally
1
symmetric spaces
1
0
2
Matthew Stover∗

c
e Universityof Michigan
D [email protected]

9
December 21, 2011
1

]
T
G Abstract
.
h Let Y be a noncompact rank one locally symmetric space of finite
t volume. Then Y has a finite number e(Y) > 0 of topological ends. In
a
m this paper, we show that for any n ∈ N, the Y with e(Y) ≤ n that are
arithmeticfallintofinitelymanycommensurability classes. Inparticular,
[
there is a constant cn such that n-cusped arithmetic orbifolds do not
1 exist in dimension greater than cn. Wemake this explicit for one-cusped
v arithmetic hyperbolicn-orbifolds and provethat none exist for n≥30.
5
9
4 1 Introduction
4
.
2 LetX bereal,complex,quaternionichyperbolicspace,ortheCayleyhyperbolic
1 plane and G its orientation-preserving isometry group. Let Γ < G be a lattice
1
and Γ\X be the associatedlocally symmetric space. Throughoutthis paper we
1
assumethatΓ\X isnoncompact,i.e.,thatΓisanonuniformlatticeinG. Then
:
v Γ\X has a finite number of topological ends e(Γ\X)> 0. The purpose of this
i
X paper is to prove the following theorem.
r Theorem 1.1. Fix X and n ∈ N. There are, up to commensurability, only
a
finitely many arithmetic lattices Γ < G such that e(Γ\X) ≤ n. Furthermore,
there is a constant c such that if dim(X)≥c , then there are no such Γ.
n n
The first statement is trivial for the hyperbolic plane, since the modular
group PSL (Z) determines the unique commensurability class of nonuniform
2
arithmetic lattices in PSL (R). For hyperbolic 3-space, the first statement is a
2
theoremofChinburg,Long,andReid [9], andfor the complexhyperbolic plane
see [18]. Since every lattice in PSp(n,1) or F(−20) is arithmetic [10, 12], the
4
arithmetic assumption is superfluous so we have the following.
∗PartiallysupportedbyNSFRTGgrantDMS0602191.
1
Corollary 1.2. For everyn>0thereare, uptocommensurability, onlyfinitely
many n-cusped quaternionic and Cayley hyperbolic orbifolds of finite volume.
We now introduce notation that we will use throughout the paper. Since G
hasrealrankone,there is aunique conjugacyclassofparabolicsubgroups. Let
P denote one such subgroup. Choose a maximal R-split torus S ⊂ P, and let
Z be the centralizer of S in G. Then there is a unipotent subgroup U ⊂ P so
that P is the semidirect product of U with Z.
Since P isthe stabilizerinGofa pointonthe idealboundaryX∞ ofX and
G acts transitively on the boundary, X∞ is naturally identified with the coset
space G/P. The ends of Γ\X are in one-to-one correspondence with the Γ-
conjugacy classes of parabolic subgroups of Γ. In other words,ends correspond
to Γ-orbits of those gP ∈ G/P such that Γ∩gPg−1 is a cocompact lattice in
gPg−1. This leads to the following interpretationofthe ends ofΓ\X when Γ is
arithmetic.
Let Γ < G be a nonuniform arithmetic lattice. Then there is an absolutely
almost simple simply connected Q-algebraic group G such that the lift of Γ to
G(R) under a central isogeny G(R) → G is commensurable with the group of
integral points G(Z) determined by a representation of G into GL (Q). See
N
[6, 5]. We describe the construction of these lattices in §2.
Since G has Q-rank one, there is a unique conjugacy class of Q-parabolic
subgroups. Let P be one, and choose a maximal Q-split torus S ⊂ P. Then Γ
acts on the complete variety (G/P) = G(Q)/P(Q). Since P(R) contains the
Q
center of G(R), we canidentify G(R)/P(R) with X∞. Parabolicsubgroupsof Γ
are commensurable with a lattice in P(Q), so
e(Γ\X)=# Γ\G(Q)/P(Q) . (1)
(cid:0) (cid:1)
The focus of this paper is on the right-hand side of (1).
We begin in §3 by studying the ends of Γ\X when X is the image in G of
a so-called principal arithmetic subgroup Γ of G(Q) defined by a coherent
Kf
compact open subgroup K of G(A ), where A denotes the finite adeles of Q.
f f f
When K is special (see [20]), then we cangive anexact formula for e(Γ \X)
f Kf
using results of Borel [4]. In other cases, we only obtain a lower bound. See
Proposition 3.4.
To prove Theorem 1.1, it suffices to consider maximal arithmetic lattices in
G. By Proposition1.4in[7],everymaximalarithmetic lattice isthe normalizer
in G of some principal arithmetic lattice. This is analyzed in §4, where we
complete the proof of Theorem 1.1. In §5, we apply our techniques to give an
explicit bound for one-cusped arithmetic hyperbolic n-orbifolds.
Theorem 1.3. One-cusped arithmetic hyperbolic n-orbifolds do not exist for
any n≥30.
It is known that there are hyperbolic reflection groups that determine one-
cusped arithmetic hyperbolic n-orbifolds for all n≤9 [13]. We close the paper
byconstructingone-cuspedhyperbolicn-orbifoldsforn=10,11. Theremaybe
2
examples for 12≤ n ≤29 related to definite rational quadratic forms with few
classes in their spinor genus; such quadratic forms do not seem to be classified,
so we do not know if Theorem 1.3 is sharp.
Acknowledgments
I thank Gopal Prasad for some helpful suggestions while I was completing this
paper.
2 Arithmetic subgroups of rank one groups
In this section, we describe the nonuniform arithmetic lattices in simply con-
nected Lie groups of R-rank one. See [19] for the full classification. This nat-
urally breaks up into three cases: hyperbolic space, complex and quaternionic
hyperbolic space, and the Cayley hyperbolic plane.
2.1 Hyperbolic space
The simply connected form of the isometry group of hyperbolic n-space is the
group Spin(n,1) [15, §27.4B], which is the double-cover of SO(n,1). Note that
we have exceptional isomorphisms
Spin(2,1)∼=SL (R),
2
Spin(3,1)∼=SL (C)
2
with the more familiar groups acting on hyperbolic 2- and 3-space.
All nonuniform arithmetic lattices in Spin(n,1) are determined as follows.
Let q be an isotropic nondegeneratequadratic form on Qn+1 of signature (n,1)
and G =Spin(q). Recall that a quadratic form is isotropic if there is a nonzero
vector v ∈ Qn+1 so that q(v) = 0. Then G(R) ∼= Spin(n,1), and every nonuni-
form arithmetic lattice in Spin(n,1) is commensurable with G(Z) for some G
as above. We note that there are constructions of arithmetic lattices for all
odd n that do not use quadratic forms, but these constructions do not lead to
nonuniform lattices. See [21] for further details.
We now describe the Q-split tori of G and their centralizers, since they are
crucialthroughoutthis paper. A maximal Q-splittorus S of G is isomorphicto
the multiplicative groupG overQ. Since q is isotropic,we canfind abasis for
m
Qn+1 such that q has matrix
0 0 1
Q= 0 Q′ 0 ,
 
1 0 0
 
where Q′ is the matrix of an anisotropic (i.e., not isotropic) quadratic form q′
onQn−1. That is, q =q ⊕q′, where q is a hyperbolic plane. Under this basis,
0 0
3
the image S ofS in the specialorthogonalgroupSO(q) is the setof matricesof
the form
x 0 0
0 In−1 0 , (2)
0 0 x−1
where x∈Q∗ and In−1 is the (n−1)×(n−1) identity matrix.
We also need to understand the centralizer Z(S) of S in G. The centralizer
Z(S) of S in SO(q) is the group of elements
x 0 0
0 A 0
0 0 x−1
 
suchthatx∈Q∗ andA∈SO(q′). ThenZ(S)istheliftofthisgrouptoSpin(q),
and the quotient of Z(S) by S is the group Pin(q′), which contains Spin(q′) as
an index 2 subgroup.
2.2 Complex and quaternionic hyperbolic space
Thesimply connectedformsofthe isometrygroupofcomplex andquaternionic
hyperbolic n-spaces are SU(n,1) and Sp(n,1), respectively [15, §24.7B]. For
complexhyperbolic space,letD be animaginaryquadraticextensionofQ. For
quaternionichyperbolicspace,letDbeadefinitequaternionalgebrawithcenter
Q, i.e., one such that D⊗ R is isomorphic to Hamilton’s quaternions H. We
Q
thenhaveaninvolutionτ :D →DgivenbythenontrivialGaloisautomorphism
when D is an imaginary quadratic field and quaternion conjugation when D is
a quaternion algebra.
Let h be an isotropic nondegenerate τ-hermitian form on Dn+1. Then h
is a τ-symmetric matrix in GL (C) or GL (H), so it has real eigenvalues.
n+1 n+1
Therefore, the signature of h makes sense, and we assume that h has signature
(n,1). If G is the special automorphism group of h, then G(R) is isomorphic to
SU(n,1) when D is imaginary quadratic and Sp(n,1) when D is a quaternion
algebra. Any nonuniform arithmetic lattice in SU(n,1) or Sp(n,1) is commen-
surable with G(Z) for some G as above. Again, there are other constructions
of cocompact lattices, but these algebraic groups suffice for the nonuniform
lattices.
We again describe some facts about centralizers of Q-split tori that we will
needlater. Asinthehyperboliccase,themaximalQ-splittorusS isisomorphic
to the multiplicative group of Q, and we can choose a basis for Dn+1 for which
0 0 1
h= 0 h′ 0 ,
 
1 0 0
 
where h′ is an anisotropic τ-hermitian form on Dn−1. Then S is realized as
matrices exactly the same as (2) and the centralizer of S now consists of those
4
matrices
x 0 0
(x,A)= 0 A 0
0 0 τ(x)−1
suchthatx∈D∗,Aisintheunitarygroupofh′ (notthespecialunitarygroup),
and xτ(x)−1det(A)=1.
We claimthatZ(S)/S is isomorphicto U(h′)(asQ-algebraicgroups)under
the map (x,A)7→A. The kernel of this map is clearly S, so it suffices to show
thatthis mapis onto. Thatis,givenA∈U(h′), we mustshowthatthereexists
x ∈ D∗ such that (x,A) ∈ SU(h). That is, we need to know that there exists
x ∈ D∗ such that x−1τ(x) = det(A). This follows immediately from Hilbert’s
Theorem 90, which holds for both imaginary quadratic fields and quaternion
algebras [15, §29.A].
2.3 The Cayley hyperbolic plane
See [1] for a more detailed description of lattices in F(−20). Let C be a Cayley
4
algebra over Q with involution τ and h be a τ-symmetric element of GL (C).
3
The automorphisms of h with reduced norm 1 form an algebraic group G that
is simply connected with G(R)∼=F(−20). One can also realize this as the auto-
4
morphisms of an exceptional Jordan algebra. The Q-split torus of G again has
the form (2), and Z(S)/S is isomorphic over Q to the group of elements in C
with reduced norm 1.
3 Principal arithmetic lattices
We begin with some generalresults. Let A be the adeles ofthe number field k
k
and A the finite adeles. We suppress the k when k=Q.
k,f
See[6]forthebasictheoryofalgebraicgroupsovernumberfields. LetGbean
absolutelyalmostsimpleandsimplyconnectedk-algebraicgroupandHbeak-
parabolicsubgroup. TheseassumptionsensurethatGhasstrongapproximation,
i.e., that G(k) is dense in G(A ) [16, Thm. 7.12]. If K ⊂ G(A ) is an open
k,f f k,f
compact subgroup, set K∞ =G(k⊗R)×K ⊂G(A). Then
f f
K∞G(k)=G(A). (3)
f
LetK beanopencompactsubgroupofG(A )andsetL =H(A )∩K .
f k,f f k,f f
Then Γ = G(k)∩K is a lattice in G(k⊗R), and we are interested in the
Kf f
quantity
eH(ΓKf)=# ΓKf\G(k)/H(k) . (4)
When k =Q andG(R) has realrank(cid:0)one, then eH(ΓKf(cid:1))=e(ΓKf\X), where X
is the symmetric space associated with G(R).
In [4, Prop. 7.5], Borel relates eH(ΓKf) to the so-called class number of H
with respect to L , which is the number
f
c(H,L )=# L∞\H(A )/H(k) . (5)
f f k
(cid:0) (cid:1)
5
Also see Chapters 5 and 8 of [16]. Since we are restating Borel’s results in dif-
ferentlanguage,andbecause one direction ofhis proofworksgreatergenerality
than his stated assumptions, we give a complete proof of [4, Prop. 7.5] in the
next two lemmas. The first step is the following general fact.
Lemma 3.1. Let G be an algebraic group over the number field k and H a
k-parabolic subgroup. Suppose that K is an open compact subgroup of G(A )
f k,f
such that K∞G(k) = G(A ). Let L = H(A )∩K and Γ = G(k)∩K .
f k f k,f f Kf f
Then
eH(ΓKf)≥c(H,Lf). (6)
Proof. Given h ,h ∈ H(A ), there exist k ,k ∈ K∞ and g ,g ∈ G(k) such
1 2 k 1 2 f 1 2
that h =k g , j =1,2. Now, suppose that there exists γ ∈Γ and h∈H(k)
j j j Kf
such that g =γg h. Then
1 2
h =k g =k γg h=(k γk−1)h h∈K∞H(A)H(k).
1 1 1 1 2 1 2 2 f
Since h (h h)−1 ∈ H(A ) and k γk−1 ∈ K∞, we must have k γk−1 ∈ L∞.
1 2 k 1 2 f 1 2 f
Therefore,if g and g are in the same Γ ,H(k) double cosetof G(k), then h
1 2 Kf 1
and h are in the same L∞,H(k) double coset of H(A ).
2 f k
Conversely, suppose h =ℓh h for some ℓ∈L∞ and h∈H(k). Then
1 2 f
g =k−1h =(k−1ℓk )g h∈K∞G(k)H(k).
1 1 1 1 2 2 f
Since g ,g ,h∈G(k), itfollowsthatk−1γk ∈Γ . Therefore,ifh andh are
1 2 1 2 Kf 1 2
in the same L∞,H(k) double coset of H(A ), then g and g are in the same
f k 1 2
Γ ,H(k) double coset of G(k).
Kf
Itfollowsthatthereisawell-definedandinjectivesetmapfromthefiniteset
L∞\H(A )/H(k) into the finite set Γ \G(k)/H(k). This proves the lemma.
f k Kf
Note that we did not use one of Borel’s assumptions: ‘G = G .H for
p op p
every p∈P’. In our language,this assumption becomes G(A )=K H(A ).
k,f f k,f
When this holds, we say that G has an Iwasawa decomposition with respect
to K and H. For example, G has an Iwasawa decomposition when K is a
f f
coherent product of parahoric subgroups and the v-adic component of K is
f
maximal and special for every nonarchimedean place v of k [20, §3.3.2]. The
following completes our proof of [4, Prop. 7.5].
Lemma 3.2. With the same assumptions and notation as Lemma 3.1, sup-
pose that G has an Iwasawa decomposition with respect to K and H. Then
f
eH(ΓKf)=c(H,Lf).
Proof. For any g ,g ∈ G(k), choose k ,k ∈ K∞ and h ,h ∈ H(A) so that
1 2 1 2 f 1 2
g = k h , j = 1,2, under the Iwasawa decomposition of G with respect to K
j j j f
andH. Notethat,bydefinitionofK∞,extendingthisfromA toA istrivial.
f k,f k
If g =γg h for some γ ∈Γ and h∈H(k), then
1 2 Kf
h =(k−1γk )h h∈K∞H(A )H(k).
1 1 2 2 f k
6
It follows that (k−1γk ) ∈ L∞ and so h and h have the same image in
1 2 f 1 2
L∞\H(A )/H(k). If h and h lie in the same L∞,H(k) double coset of
f k 1 2 f
H(A ), a similar computation shows that g and g have the same image in
k 1 2
Γ \G(k)/H(k). This proves that
Kf
eH(ΓKf)≤c(H,Lf),
so the two are equal by Lemma 3.1.
We also need the following analogue of [4, Prop. 2.4].
Lemma 3.3. Suppose that
π
1→C →G →H→1
is a central exact sequence of k-algebraic groups. Let K ⊂G(A ) be an open
f k,f
compact subgroup and L = π(K ). Then c(G,K ) ≥ c(H,L ). Moreover, if
f f f f
c(C,C(A )∩K )=1 then c(G,K )=c(H,L ).
k,f f f f
Proof. By assumption, we have a natural surjective map
πˆ :K∞\G(A )/G(k)→L∞\H(A )/H(k).
f k f k
The first statement follows immediately.
Wemustshowthatπˆ isinjectivewhenc(C,C(A )∩K )=1. Supposethat
k,f f
g ,g ∈ G(A ) and π(g ) = xπ(g )y for some x ∈ L∞ and y ∈ H(k). Then we
1 2 k 1 2 f
have x˜∈K∞ and y˜∈G(k) such that g−1x˜g y˜∈C(A ). However,
f 1 2 k
C(A )= C(A )∩K∞ C(k),
k k f
so there exist x1 ∈C(Ak)∩Kf∞ an(cid:0)d y1 ∈C(k) s(cid:1)uch that
g−1x˜g y˜=x y .
1 2 1 1
Since C is central, we get
g =(x−1x)g (yy−1)∈K∞g G(k).
1 1 2 1 f 2
Thus πˆ is injective.
Now, we return to the case where k =Q and G(R) is rank one. Let P =H
be a Q-parabolic subgroup of G. It is unique up to conjugacy [6, Prop. 21.12].
LetS ⊂P be a maximal Q-splittorus. ThenP is a semidirect productZ(S)U,
where Z(S) is the centralizer of S in G and U is unipotent.
WecallanopencompactsubgroupK ⊂G(A )coherent ifitisdefinedbya
f f
coherentcollectionofparahoricsubgroupsofG(Q )forallp;see[7]orAppendix
p
A of [16]. Let G be the orientationpreserving isometry groupof the symmetric
space X and Γ < G be a nonuniform lattice. We say that Γ is a principal
arithmetic lattice if there is an absolutely almost simple and simply connected
Q-group G and a coherent open compact subgroup K ⊂ G(A ) such that Γ is
f f
the image in G of Γ = K ∩G(Q) under a central isogeny G(R) → G. The
Kf f
following allows us to further refine the conclusions of Lemma 3.2 for principal
arithmetic lattices.
7
Proposition 3.4. Let G be a Q algebraic group of real and Q-rank one and P
be a Q-parabolic subgroup with S ⊂ P a maximal Q-split torus and Z(S) the
centralizer of S in G. Let K ⊂G(A ) be an open compact subgroup determined
f f
by a coherent product of parahoric subgroups. Set:
L =K ∩H(A ),
f f f
M =K ∩Z(S)(A ),
f f f
Γ =K ∩G(Q).
Kf f
If X is the symmetric space for G(R), then
e(Γ \X)≥c(Z(S),M )=c(H,Mˆ ), (7)
Kf f f
where H = Z(S)/S and Mˆ is the image of M in H. Moreover, we have
f f
equality in (7) when G has an Iwasawa decomposition with respect to K and
f
P.
Proof. We will prove that
c(P,L )=c(Z(S),M ).
f f
Since the torus S is the multiplicative group over Q, it has class number one.
Therefore, the right-hand equality in (7) follows from Lemma 3.3. The propo-
sition then follows immediately from Lemmas 3.1 and 3.2.
Let U be the unipotent Q-group such that P = Z(S)U. By Corollary 2.5
and Proposition 2.7 in [6], it suffices to show that
∞ ∞ ∞
L =M N ,
f f f
where N∞ = U(R) × (K ∩ U(A )), and it suffices to prove the analogous
f f f
decomposition at any nonarchimedeanplace. However,when the component of
K atafixednonarchimedeanplaceisaparahoricsubgroup,thisdecomposition
f
follows immediately from the italicized statement in [20, §3.1.1]. This proves
the proposition.
Lastly,wewillneedthe followingrelationshipbetweenclassnumbersofuni-
taryandspecialunitarygroups. Thisresultmaybeknowningreatergenerality,
but we could not find a reference.
Proposition 3.5. Let D be Q, an imaginary quadratic field, or a definite
quaternion algebra over Q, and let τ be trivial, the nontrivial Galois automor-
phism, or quaternion conjugation, respectively. Let h be a τ-hermitian form on
DN, H the pin/unitary group of h, and H ⊂H the spin/special unitary group.
0
Let M ⊂ H(A ) be an open compact subgroup, M∞ = H(R)×M ⊂ H(A),
f f f f
and L∞ =H (A)∩M∞. Then there is a universal constant c so that
f 0 f
1
# M∞\H(A)/H(Q) ≥ # L∞\H (A)/H (Q) (8)
f c f 0 0
(cid:0) (cid:1) (cid:0) (cid:1)
8
Proof. It suffices to show that for any g ∈ H (A), the elements of H (A) in
0 0
the double coset M∞gH(Q) project to at most c elements of L∞\H(A)/H(Q).
f f
Suppose that g = kgh for g ∈ H (A), k ∈ M∞, and h ∈ H(Q). Since H (A)
1 1 0 f 0
is the kernel of the determinant map
d:H(A)→D∗(A),
we have d(k) =d(h)−1. The image of d is contained in the subgroup D1(A) of
elements in D(A) of reduced τ-norm 1. Since d(M∞) lies in an open compact
f
subgroup of D1(A) and d(H(Q)) lies in the rational points, it follows that d(k)
and d(h) must lie in d(M∞)∩d(H(Q)), which is contained in the subgroupO1
f
of reduced τ-norm 1 elements of some Z-order O of D(Q).
Then O1 is a finite group of order bounded by a universal constant c .
0
Indeed, c = 2 if D = Q, is 6 is D is imaginary quadratic, and is 24 when D
0
is a definite quaternion algebra. Therefore, we can choose elements r ,...,r
1 c0
in M∞ and s ,...,s ∈H(Q) so that if k ∈M∞ and d(k) =d(r ) then there
f 1 c0 f j
exists ℓ∈L∞ so thatk =ℓr andif d(h)=d(s )for some h∈H(Q)then there
f j j
exists h ∈H (Q) so that h=s h .
0 0 j 0
Therefore, if g ∈ H (A) and g = kgh for some k ∈ M∞ and h ∈ H(Q),
1 0 1 f
there exist r , s , ℓ∈L∞, and h ∈H (Q) such that
i j f 0 0
g =ℓ(r gs )h .
1 i j 0
Then r gs ∈H (A), so g has the same image in L∞\H (A)/H (Q) as one of
i j 0 1 f 0 0
the elements of the finite set {r gs }. There are at most c = c2 such elements,
i j 0
so this proves the proposition.
Nowweexplaintheseresultsingreaterdetailforrealandcomplexhyperbolic
space. In particular, we give sufficient information to compute the number of
cusps for principal arithmetic subgroups of interest.
Hyperbolicspace. Recallfrom§2.1,H isthespingroupofq′whereq =q ⊕q′
0 0
forq ahyperbolicplane. Asexplainedin[4]fortheorthogonalgroup,theclass
0
number of H is the number of classes in the spinor genus of q′ with respect to
0
the lattice L′, where L′ is the summand of L associated with q′. In particular,
we see that Theorem 1.1 follows for principal arithmetic lattices from the fact
that there are only finitely many anisotropic quadratic forms over Q with at
most n classes in its spinor genus [8, Appendix A.3].
FortheBianchigroupsSL (O ),thenumberofcuspsiswell-knowntoequal
2 k
the class number h of the imaginary quadratic field k. See [9] for a proof.
k
The quadratic form over Q that determines the corresponding subgroup of
Spin(3,1) ∼= SL (C) is the binary hermitian form of discriminant equal to the
2
discriminant of the imaginary field k. In particular, the above methods show
that SL (O ) has h cusps via Gauss’s work on binary quadratic forms.
2 k k
Complex hyperbolic space. Let ℓ be an imaginary quadratic field and h a
hermitian form of signature (n,1) on the ℓ-vector space V of dimension n+1.
9
Then G is the special unitary group of h and H is the unitary group of the
anisotropic summand h′. Therefore, the number of cusps correspond precisely
to the class number of the unitary group of a hermitian lattice. This is closely
related to the class number and restricted class number of imaginary quadratic
fields. See [18] for a complete analysis in the case n=2 and [24, 14] for special
cases in higher dimensions.
Quaternionichyperbolicspace hasa similardescriptioninterms ofa hermi-
tian formon a definite quaternionalgebraoverQ. We leave it to the interested
reader to work out this case and the Cayley hyperbolic plane in further detail.
We now proceed to maximal lattices and the proof of Theorem 1.1.
4 Maximal lattices
LetG be a simply connectedQ-algebraicgroupofrealandQ-rankone as in §3,
and let Λ < G(R) be a maximal lattice. To prove Theorem 1.1, we must show
that for every x ∈ N, there are only finitely many G such that e(Γ\X) ≤ x,
where X is the symmetric space associated with G. From [7, Prop. 1.4], we
know that there exists a coherent open compact subgroup K ⊂ G(A ) such
f f
that Λ is the normalizer in G(R) of Γ =G(Q)∩K .
Kf f
Let S be a maximal Q-split torus of G, and H = Z(S)/S be the quotient
of the centralizer of S in G by S. Let H be the subgroup of H consisting of
0
elements with determinant one. It follows from Propositions 3.4 and 3.5 that
e(Λ\X)≥ e(ΓKf\X) ≥ 1c(H0,Mˆf), (9)
[Λ:Γ ] c [Λ:Γ ]
Kf Kf
where Mˆ is the open compact subgroup of H (A ) determined by K and c
f 0 f f
is the constant from Proposition 3.5. Also, recall from §3 that H uniquely
0
determines G. Therefore, it suffices to show that the right-hand side of (9) is
bounded above by x for only finitely many H .
0
Let C be the center of G. Then [7, Prop. 2.9] gives an exact sequence
1→C(R)/(C(Q)∩Γ )→Λ/Γ →δ(G(Q))′ →1,
Kf Kf Θ
where G is the adjoint form of G and δ(G(Q))′ is the image of Λ in H1(Q,C).
Θ
The central elements of Λ clearly act trivially on X∞, so we in fact have
1 c(H ,Mˆ )
0 f
e(Λ\X)≥ . (10)
c#δ(G(Q))′
Θ
Once we prove that the right-hand side of (10) is bounded above by x for only
finitely many H , the proof of Theorem 1.1 will be complete. In other words,
0
we need to prove the following.
Theorem 4.1. Let G be a Q-algebraic group of real and Q-rank one. Choose
a maximal Q-split torus S in G, and let Z(S) be the centralizer of S in G. Let
10