ContemporaryVMathematicsolume 00, 0000
Buildings, Group Extensions and the
Cohomology of Congruence Subgroups
ALEJANDRO ADEM
Introduction
Let Tn denote the Tits Building associated to the discrete group SL n(Z).
In this note we will be interested in computing the equivariant cohomology
H* (|Tn |; Z), where |Tn | is the geometric realization of Tn , and is a torsi*
*on-
free discrete subgroup of finite index in SL n(Z). In the special case n = 3, *
*we
will see that this equivariant cohomology in fact corresponds to the ordinary c*
*o-
homology of a 4-dimensional Poincare Duality group which has as its quotient.
This is then applied to partially compute the cohomology of the level p con-
gruence subgroups (p an odd prime) in SL 3(Z). In particular we obtain
Theorem. Let (p) denote the level p congruence subgroup of SL 3(Z); then
dim Q H3((p); Q) _1_12(p3 - 1)(p3 - 3p2 - p + 15) + 1:
It turns out that this method can be used for other rank two groups such as
Sp 4(Z) and G 2(Z). In x6 we apply this to the level p congruence subgroups in
Sp 4(Z), obtaining the cohomology of the relevant parabolic subgroups and from
this a lower bound for the fourth betti number:
Theorem. Let (p) denote the level p congruence subgroup in Sp 4(Z); then
dim Q H4((p); Q) _1_24(p4 - 1)(2p3 - 3p2 - 2p + 27) + 1:
Our methods are mostly algebraic, involving techniques from group extensions
and their cohomology. We offer a systematic approach which works with general
coefficients and in addition we provide complete information on the cohomology
of the parabolic subgroups. One of the key facts which we prove is that the
______________
1991 Mathematics Subject Classification. Primary 20J Secondary 55R.
Partially supported by an NSF Grant
cO0000 American Mathematical*
* Society
0000-0000/00 $1.00 + $.25*
* per page
1
2 ALEJANDRO ADEM
congruence subgroups are homomorphic images of Poincare duality groups with
kernel a free, infinitely generated group. These groups are interesting in the*
*ir
own right, and probably deserve further attention. They can be realized as the
fundamental groups of the Borel-Serre compactification; this is discussed in x7.
The application to the rational cohomology of congruence subgroups in SL3 (Z)
is contained in the paper by Lee and Schwermer (see [LS]), although it is perha*
*ps
not as well known as it should be. Similarly there is overlap for Sp4(Z) (see
[S]). In fact most of the work in this paper was done without knowledge of the
existence of these papers. We recommend them to the reader.
In some sense this is an expository paper. One of its goals is to put com-
putations for congruence subgroups within a simplified framework of group co-
homology. Complete answers involve assembling a wide array of mathematical
techniques beyond our scope, but at the very least one hopes this paper will
motivate the reader to learn the more number-theoretic aspects of the subject
(see [A], [AGG]).
The author is grateful to A. Ash and L. Solomon for helpful comments.
x0. Preliminaries
We recall that SL n(Z) is a group of virtual cohomological dimension equal
to N = n(n-1)_2. The module known as the Steinberg module St is a dualizing
module for any torsion-free subgroup of finite index in SL n(Z); that is there
are isomorphisms
Hi(; M ) ~=HN-i (; St M )
where M is any Z module, i any integer and these equivalences are given
by taking cap product with a fixed element z 2 HN (; St); dually, we have
isomorphisms
Hi(; M ) ~=HN-i (; Hom (St; M )):
Recall that SL n(Q) acts naturally on the set of all proper non-trivial subs*
*paces
of Qn . This defines a partially ordered set Tn and hence its geometric realiza*
*tion
|Tn | which can be thought of as an SL n(Z)-CW complex of dimension n - 2.
Furthermore we have
8
>>>Z *=0
>>>
<
H*(|Tn |; Z) ~=> St *=n-2
>>>
>>:
0 otherwise.
For details on these facts we refer to [Br1] and [Br2].
Next we recall the notion of equivariant cohomology: if G is a group acting
on a space X, then
H*G(X) = H*(X xG EG; Z)
COHOMOLOGY OF CONGRUENCE SUBGROUPS 3
where EG is the universal free contractible G space and X xG EG = X x EG=G
(diagonal action).
We also recall that if X is a G-CW complex, then there exist two spectral
sequences for computing H*G(X), namely
(I) Ep;q2= Hp(G; Hq(X; Z)) ) Hp+qG(X)
and
(II) Ep;q1= Hq(G; Cp(X)) ) Hp+qG(X)
where C*(X) denotes the module of cellular p-cochains on X.
We remark here for later use that if
M
Cp(X) ~= Z[G=Goei];
i2I
then M
Hq(G; Cp(X)) ~= Hq(Goei; Z):
i2I
x1. The Equivariant Cohomology of |Tn |.
We will prove the following
Theorem 1.1. Let SL n(Z) denote a torsion free subgroup of finite index,
then there is a duality isomorphism
Hi (|Tn |) ~=H n2+n-4_ (|Tn |):
2 -i
Proof. To begin we observe that there is a fibration
|Tn | -! |Tn |?x E
?y
B
where dim |Tn | = n - 2, dim B = N = n(n-1)_2and hence dim |Tn | x E =
2+n-4
n - 2 + N = n______2. Let us denote this number by M . Now in this case the
spectral sequence of type (I) in homology degenerates to a long exact sequence
. . .! Hi-n+2 (; St) ! Hi (|Tn |) ! Hi(; Z) ! Hi-n+1 (; St) ! . .:.
Note in particular that
HM (|Tn |) ~=HN (; St) ~=H0(; Z) ~=Z:
Hence there is a "fundamental class" 2 HM (|Tn |) which can be identified wi*
*th
the class z 2 HN (; St) which induces the duality isomorphisms Hi(; M ) ~=
HN-i (; St M ) via the cap product. We claim that
\ : Hi (|Tn |) ! HM-i (|Tn |)
induces a duality isomorphism.
4 ALEJANDRO ADEM
To prove this we look at the long exact sequence in cohomology:
. ! Hi(; Z) ! Hi (|Tn |) ! Hi-n+2 (; St*) ! Hi+1 (; Z) ! . .:.
Cap product with induces maps from one long exact sequence to the other:
Hi(;?Z) ! Hi?(|Tn |) ! Hi-n+2?(; St*) ! Hi+1?(; Z)
?y ~= ?y \ ?y ~= ?y ~=
HN-i (; St) ! HM-i (|Tn |) ! HM-i (; Z) ! HN-i-1 (; St)
Applying the five-lemma completes the proof.
Consider the special case when n = 3; then |T3| is an infinite graph, with f*
*ree
fundamental group F . In this case |Tn | x E is aspherical and 4 dimensional;
hence if Q = ss1(|Tn | x E) we have an extension
1 ! F ! Q ! ! 1
where Q is a 4-dimensional Poincare Duality group with homology of finite type.
In particular we have a long exact sequence:
0 ! H1(; Z) ! H1(Q; Z) ! H0(; St*) ! H2(; Z) ! H2(Q; Z)
! H1(; St*) ! H3(; Z) ! H3(Q; Z) ! H2(; St*)
! H4(; Z) ! H4(Q; Z) ! H3(; St*) ! 0
which simplifies to yield
(1:2)
0 ! H1(; Z) ! H1(Q; Z) ! H3(; Z) ! H2(; Z) ! H2(Q; Z) !
! H2(; Z) ! H3(; Z) ! H3(Q; Z) ! H1(; Z) ! 0
and H4(Q; Z) ~=Z.
x2. Cohomology of Parabolic Subgroups
From now on we specialize to the case n = 3 and = (p), a level p con-
gruence subgroup, p an odd prime and Q = Q(p). We will compute the integral
cohomology of its intersections with the parabolic subgroups in SL 3(Z).
To begin let B SL n(Z) denote the subgroup of upper triangular matrices
in SL 3(Z). Then
8 0 1 9
< 1 c1 c2 =
B \ (p) = : @ 0 1 c3 A p divides c1, c2, and c3 :
0 0 1 ;
COHOMOLOGY OF CONGRUENCE SUBGROUPS 5
The following elements generate this group
0 1 0 1 0 1
1 p 0 1 0 0 1 0 p
a = @ 0 1 0 A ; b = @ 0 1 p A ; c = @ 0 1 0 A
0 0 1 0 0 1 0 0 1
and aba-1 b-1 = cp.
In fact we may express B \ (p) as a central extension
1 ! Z ! B \ (p) ! Z Z ! 1
_
where is the central subgroup and __a, b generate the quotient. We have an
exact sequence in cohomology (over Z):
0 ! Z Z ! H1(B \ (p)) ! Z-d2!Z ! H2(B \ (p)) ! Z Z ! 0
and H3(B \ (p)) ~=Z. From the extension data, coker d2 ~=Z=p and hence we
have proved
Proposition 2.1.
8
>>>Z i=0,3
>>>
>>>
>>
>>>
>>>Z Z Z=p i=2
>>>
>:
0 otherwise:
This is a 3-dimensional Poincare Duality group. Next we consider the par-
abolic subgroup P1 SL 3(Z) the subgroup with zeros in the second and third
entries of its first column. We have
8 0 1 fi
>>< 1 x y fifi oe
B 0 C fip divides x and y
P1 \ (p) = > B@ CA fifi :
>: A fi A 2 2(p)
0 fi
Let 8 0 1 9
< 1 x y =
E = : @ 0 1 0 A p divides x and y ;
0 0 1 ;
1 . . .
then E ~=Z Z and if we embed 2(p) in (p) via A 7! .. ____ we obtain a
. |_A_|
semidirect product decomposition
___________________________
| P1 \ (p) = E xT 2(p): |
|__________________________|
6 ALEJANDRO ADEM
If A = ac db 2 2(p), the action on E is given by
0 1 0 1
1 x y 1 ax + cy bx + dy
@ 0 1 0A TA7!@0 1 0 A
0 0 1 0 0 1
or
x 7! ax + cy
y 7! bx + dy
and hence
TA xy = ab dc xy = AT xy :
We can now compute H*(P \ (p); Z); we use the Lyndon-Hochschild-Serre
spectral sequence for the extension above. It has E2 term
Ep;q2= Hp(2(p); Hq(E; Z)):
Note that as 2(p) is free (and hence 1-dimensional), there are no differentials
and it collapses at E2. We obtain
8
>>>Z i=0
>>>
>>>
>>#
>>> 1 1 2 2(p)
>>>H (2(p); H (E; Z)) H (E; Z) i=2
>>>
>:
H1(2(p); H2(E); Z) i=3:
Now the 2(p) action on H1(E) is the natural action on M = Z Z, and
hence that on H2(E) is via the determinant, from which H1(E; Z)2(p) = 0,
H2(E; Z)2(p) ~= Z and we have proved
Proposition 2.2.
8
>>>Z i=0
>>>
>>>
>>>H1( (p); Z) i=1
>>> 2
>><
Hi(P1 \ (p); Z) ~=> H1(2(p); M ) Z i=2
>>>
>>>
>>> 1
>>>H (2(p); Z) i=3
>>>
>:
0 otherwise:
COHOMOLOGY OF CONGRUENCE SUBGROUPS 7
Denote by M * the dual 2-dimensional representation of 2(p). Then it is
direct to verify that for the other parabolic subgroup P2, we have
Proposition 2.3.
8
>>>Z i=0
>>>
>>>
>>>H1( (p); Z) i=1
>>> 2
>><
Hi(P2 \ (p); Z) ~=> H1(2(p); M *) Z i=2
>>>
>>>
>>> 1
>>>H (2(p); Z) i=3
>>>
>:
0 otherwise:
Note that 2(p) is a free group of rank 1 + (p-1)p(p+1)_12:
x3. Double Cosets and the Tits Building
We recall that |T3| is a 1-dimensional SL 3(Z)-complex with 2 orbits of zero
cells, and one orbit of 1-cells.* The respective isotropy subgroups are P1, P2 *
*and
B.
From the usual induction-restriction formula, we have
M
Z[SL 3(Z)=B]| ~= Z[= \ xiBx-1i]
xi2I0
where I0 = \SL 3(Z)=B is the set of double cosets. Similarly we have
M
Z[SL 3(Z)=P1]| ~= Z[= \ yjP1y-1j]
yj2I1
M
Z[SL 3(Z)=P2]| ~= Z[= \ zkP2z-1k]
zk2I2
where I1 = \SL 3(Z)=P1, I2 = \SL 3(Z)=P2.
Note that as / SL3(Z), we have \ xiBx-1i~= \ B, \ yjP1y-1j~= \ P1,
\ zkP2z-1k~= \ P2, for all i; j and k.
______________*
By the `Invariant Factor Theorem', SL3(Z) acts transitively on one and two *
*dimensional
subspaces in Q3, as well as on the flags.
8 ALEJANDRO ADEM
Next note that the cardinality of the indexing can be determined as follows**
**:
#I0 = [SL 3(Z)= : B= \ B] = [SL 3(p) : B(p)]
#I1 = [SL 3(Z)= : P1= \ P1] = [SL 3(p) : P1(p)]
#I2 = [SL 3(Z)= : P2= \ P2] = [SL 3(p) : P2(p)]
where B(p), P1(p), P2(p) denote the corresponding subgroups in SL 3(p).
The orders of these subgroups can easily be determined to be
|B(p)| = 4p3 and |P1(p)| = |P2(p)| = 2p3(p2 - 1):
From this we readily deduce that
3 - 1)(p2 - 1) p3 - 1
#I0 = (p______________4 and #I1 = #I2 = _______2:
We now use this to describe the cellular chains on |T3| as -modules. Given
that over SL 3(Z) we have
C0(|T3|)~= Z[SL 3(Z)=P1] Z[SL 3(Z)=P2]
C1(|T3|)~= Z[SL 3(Z)=B]
we obtain that as -modules:
0 p3-1 1 0 1
____2M p3-1_2M
C0(|T3|) ~=B@ Z[= \ yjP1y-1j]CA B@ Z[= \ zkP2z-1k]CA
(p3-1)(p2-1)_4
M
C1(|T3|) ~= Z[= \ xiBx-1i]:
We note here that dividing out by the -action we can describe |T3|= in
terms of the mod p building associated to SL 3(p). This a (p + 1)-valent graph
which can be described explicitly.
______________**
These indexing sets will also correspond to the number of -conjugacy class*
*es of parabolic
subgroups.
COHOMOLOGY OF CONGRUENCE SUBGROUPS 9
x4. Cohomology Calculations
We now use the second spectral sequence described in x0, applied to the 1-
dimensional -complex |T3|; we obtain a long exact sequence
. . .! Hi (|T3|) ! Hi(; C0(|T3|)) ! Hi(; C0(|T3|)) ! Hi+1 (|T3|) ! . . .
Combining the results in x2, x3, we have
0 p3-1 1 0 1
____2M p3-1_2M
H*(; C0(|T3|)) ~= B@ H*(P1 \ (p); Z)CA B@ H*(P2 \ (p); Z)CA
(p3-1)(p2-1)_4
M
H*(; C1(|T3|)) ~= H*(B \ (p); Z):
We are now ready to substitute explicit values:
3-1 (p3-1)(p2-1)_ 1 [12+(p-1)p(p+1)_]hp3-1_i
0 ! Z = H0 (|T3|) ! Zp ! Z 4 ! H (|T3|) ! Z 6 2 !
(4.1) 2 p3-1 3
____
(p3-1)(p2-1)_ 2 6 M 2 1 * 7
Z 2 ! H (|T3|) ! 4 H (2(p); M M ) Z Z5 !
2 (p3-1)(p2-1) 3
___________4 p3-1 (p3-1)(p2-1)
64 M Z Z Z=p75 ! H3 M [12+(p-1)p(p+1)_12] ___________4
(|T3|) ! Z ! Z *
* !
H4 (|T3|) ! 0:
We now derive some consequences from this sequence. Recall first that O(B(p)*
*) =
0 and hence from (1.2) we deduce O(BQ(p)) = O(|T3| x E) = 0: Hence if F is
a field of characteristic prime to p, we infer from (4.1) that
dim FH1(2(p); MF MF*) = (p_-_1)p(p_+_1)_3
where MF is the 2-dimensional natural representation of 2(p). In addition
one can verify that if (p; q) = 1, then H1(2(p); M M *) is q-torsion-free. This
follows from looking at the long exact sequence associated to mod q reduction a*
*nd
the fact that if F is a field of characteristic prime to p, then (MF MF*)2(p) *
*= 0.
Reducing mod p, we obtain a long exact sequence of the form
(p-1)p(p+1)_
0 ! (Fp)4 ! H1(2(p); M M *)-.p!H1(2(p); M M *) ! [Fp]4(1+ 12 ) ! 0:
10 ALEJANDRO ADEM
Observing that [(M M *) Z=p2]2(p) ~= (Z=p)4, we infer (now using the mod
p2 reduction sequence) that H1(2(p); M M *) has no elements of order p2 and
so we deduce
Proposition 4.2.
(p-1)p(p+1)_ 4
H1(2(p); M M *) ~=[Z] 3 (Z=p) :
Let fi3 denote the third betti number associated to the classifying space of*
* a
group. Looking at the final part of the sequence (4.1), we find that
3 2
fi3(Q(p)) (p3 - 1) 1 + (p_-_1)p(p_+_1)_12- (p__-_1)(p__-_1)_4+ 1
which, combined with (1.2) and the fact that H1(; Z) = 0 (see [K] or [Se], pg.
122) yields
Theorem 4.3.
fi3((p)) fi3(Q(p)) _1_12(p3 - 1)(p3 - 3p2 - p + 15) + 1:
x5. The Symplectic Group: Parabolic Subgroups
We will use the methods previously described to analyze the cohomology of
the congruence subgroups in Sp 4(Z). The key fact here is that Sp 4(Z) is also a
rank two group, hence its Tits Building will once again be a graph with an edge
transitive action of the group. As before, the cohomology can be approached
using the parabolic subgroups.
We proceed as before, now let |W | denote the geometric realization of the
Tits Building for Sp 4(Z); we denote L(p) = ss1(|W | x(p) E(p)), where now
(p) Sp4(Z) is a level p congruence subgroup, p an odd prime. In this instance,
(p) has cohomological dimension equal to 4, and hence L(p) has cohomological
dimension equal to 5; as before it will be a Poincare Duality group.
We need to describe the parabolic subgroups in some detail. If
0 0 0 0 -1 1
B 0 0 -1 0 C
J = B@0 1 0 0 CA
1 0 0 0
then Sp4(Z) can be described as the matrices A 2 SL 4(Z) such that AT JA = J.
It is convenient to write
A = A1A A2
3 A4
where A1; A2; A3; A4 are 2 x 2 matrices. Then, if we write
J = 0Q -Q0 ;
COHOMOLOGY OF CONGRUENCE SUBGROUPS 11
the conditions on A can be rewritten as
AT3QA1 - AT1QA3 = 0; A3QA2 - AT1QA4 = -Q
AT4QA1 - AT2QA3 = Q; AT4QA2 - AT2QA4 = 0:
The parabolic subgroups in Sp 4(Z) can be obtained by intersecting with ap-
propriate parabolics in SL 4(Z). To begin we consider the parabolic subgroup G1
of symplectic matrices with A3 = 0. Manipulating the defining equations one
can easily see that if
S = I0 Q0 ;
then the conjugate group SG0S-1 can be described as the set of matrices
D B
0 (DT )-1
where D 2 GL 2(Z) and BDT = DBT .
Next we consider the symplectic matrices where the first column has zero
entries except in the top corner. To simplify things we will assume that this
non-zero entry is 1. After some manipulation we obtain a description of this
group (denoted G2) as the matrices of the form
0 0 1 1
1 (u1 u2) C w
BB -1 0 C
B@0 c11 c12 u1CC
0 c21 c22 u2A
0 0 0 1
where
C = c11c c12 2 SL 2(Z); u1; u2; w 2 Z:
21c22
Letting C 2 GL 2(Z) we obtain an index 2 extension group which is evidently a
maximal parabolic, denoted G2.
The intersection G0 = G1 \ G2 can then be described as the determinant
one matrices of the form
0 1 x y z 1
BB 0 1 w y - xw CC
@ 0 0 1 -x A
0 0 0 1
where x; y; z; w 2 Z. As expected, this coincides with the intersection UT \
Sp 4(Z), where UT is the subgroup of all 4x4 matrices which are upper triangula*
*r.
We will not dwell on the parabolics themselves, rather we are interested in the*
*ir
intersections with the congruence subgroup (p).
12 ALEJANDRO ADEM
5.1 The group G1 \ (p).
Using the fact that (p) / Sp4(Z), this group can be described as the matrices
of the form
D B
0 (DT )-1
where D 2 2(p) SL 2(Z), B = pE and EDT = DET . Let V denote the
subgroup consisting of all matrices in G1 \ (p) of the form
I B
0 I ;
which one can easily check is a normal subgroup isomorphic to Z3. Using this it
follows that
G0 \ (p) ~=V xT 2(p)
where X 2 2(p) acts via
I B I XBXT
0 I 7! 0 I :
In fact this action can be expressed more generally as the restriction of a hom*
*o-
morphism T : SL 2(Z) ! SL 3(Z) given by
0 a2 2ab b2 1
a b @ A
c d 7! acc2 ad2+cbcd bdd2 :
5.2 The group G2 \ (p).
In the description of G2 given above, we simply require that C 2 2(p)
SL 2(Z), u1; u2; z 2 pZ and that the other 2 diagonal entries be equal to one.
Let U denote the normal subgroup consisting of matrices in G2 \ (p) such that
C = I. Then we can write the group as a semidirect product
U xT 2(p)
where an element C 2 2(p) acts via
0 0 1 1 0 0 1 1
1 (u1 u2) z 1 (u1 u2) C-1 z
BB -1 0 C B -1 0 C
B@ 0 I u1 CC 7! BB0 I C u1 *
*CC
u2 A @ u2 A
0 0 1 0 0 1
5.3 The group G0 \ (p).
We conjugate to get this into suitable form:
0 1 0 1
1 x y z 1 x z y
I 0 BB0 1 w y - xw CC I 0 BB0 1 y - xw w CC
0 Q @ 0 0 1 -x A 0 Q = @ 0 0 1 0 A
0 0 0 1 0 0 -x 1
COHOMOLOGY OF CONGRUENCE SUBGROUPS 13
from which we deduce that
G0 \ (p) ~=V xT Z
where the group Z is generated by
0 1 2p p2 1
t = 10 p1 ; T (t) = @ 0 1 p A :
0 0 1
x6. The Symplectic Group: Cohomology Calculations
We are now in a position to calculate the cohomology of the groups Gi\ (p).
Using the spectral sequence associated to the extension in 5.1, we have
Theorem 6.1.
8
>>>Z i=0
>>>
>>> 1+(p-1)p(p+1)
>>>(Z) ___________12 i=1
>>>
>>>
>>> 1 1
>< H (2(p); H (V; Z)) i=2
Hi(G1 \ (p); Z) ~=>
>>> 1 2
>>>Z H (2(p); H (V; Z)) i=3
>>>
>>> (p-1)p(p+1)
>>>(Z)1+ __________12 i=4:
>>>
>>>
:
0 otherwise.
To compute H*(G2 \ (p); Z), we first observe that
8
>>>Z i=0
>>>
>>>
>>
>>>
>>>Z Z Z=p i=2
>>>
>:
Z i=3:
As a 2(p)-module, the cohomology of H*(U; Z) is trivial in dimensions 0 and 3;
the dual of the 2-dimensional natural representation M *in H1; and M * Z=p,
where Z=p has a trivial action in H2. From this information we can obtain
14 ALEJANDRO ADEM
Theorem 6.2.
8
>>>Z i=0
>>>
>>>
>>>H1(2(p); Z) i=1
>>>
>>>
>>> 1 *
>< H (2(p); M ) Z=p i=2
Hi(G2 \ (p); Z) ~=>
>>> 1 * 1
>>>Z H (2(p); M ) H (2(p); Z=p) i=3
>>>
>>> (p-1)p(p+1)
>>>(Z)1+ __________12 i=4
>>>
>>>
:
0 otherwise.
To compute H*(G0 \ (p); Z) we use the semidirect product description pro-
vided in 5.3 to obtain
H*(G0 \ (p); Z) ~=H0(Z; H*(V; Z)) H1(Z; H*-1 (V; Z)):
These terms can be explicitly computed, yielding
Theorem 6.3.
8
>>>Z i=0
>>>
>>>
>>>Z Z i=1
>>>
>>>
>>>
>< Z Z Z=p Z=2p i=2
Hi(G0 \ (p); Z) ~=>
>>>
>>>Z Z Z=p Z=2p i=3
>>>
>>>
>>>Z i=4.
>>>
>>>
:
0 otherwise.
Just as before, the cohomology of these subgroups can be used to estimate
the cohomology of (p). We will concentrate here on finding a lower bound for
fi4((p)). For this we need to compute the indices ji = [Sp 4(Fp) : Gi(p)] where
COHOMOLOGY OF CONGRUENCE SUBGROUPS 15
Gi(p) denote the corresponding groups reduced mod p. The following can readily
be established:
|Sp 4(Fp)| = p4(p4 - 1)(p2 - 1); |G0(p)| = 8p4; |G1(p)| = 2p4(p2 - 1) = |G2(p)*
*|:
From this we deduce
j0 = (p4 - 1)(p2 - 1)=8; j1 = j2 = (p4 - 1)=2:
As in x4 we have a long exact sequence which can be used to estimate the
cohomology of (p). In this case when we look at the end of the sequence we
obtain a lower bound for fi4(L(p)) and hence we have
Theorem 6.4.
fi4((p)) fi4(L(p)) _1_24(p4 - 1)(2p3 - 3p2 - 2p + 27) + 1:
Note that we have a complete integral calculation for the cohomology of the
constituent subgroups. Hence a calculation over the field Fp is also possible. *
*Also
it is interesting to note the presence of 2-torsion in the cohomology of G0 \ (*
*p).
x7. Final Remarks
Let A = P1 *B P2 denote the usual amalgamated product of parabolics in
SL 3(Z). This group acts in the usual way on an infinite tree K with a single
edge as a quotient. Now there is a natural surjection A ! SL 3(Z) induced by
the inclusions of the parabolic subgroups. If the kernel of this map is denoted*
* by
F , then it is the fundamental group of an infinite graph on which SL 3(Z) acts
with quotient a single edge. In fact this graph can be identified with the Tits
building, and from our previous constructions it is easy to see that the group
Q(p) is a finite index subgroup fitting into an extension
1 ! Q(p) ! A ! SL 3(Fp) ! 1:
In other words, the amalgam A is a 4-dimensional virtual duality group, mapping
onto SL 3(Z). In fact this is a purely algebraic construction of the Borel-Ser*
*re_
compactification for SL 3(Z)._More precisely, it is easy to verify that_if_X i*
*s their
construction, then ss1(@(X =(p))) ~=Q(p). Observe that in this case @(X =(p))
is a closed 4-dimensional manifold, in fact homotopy equivalent to the Borel
construction |T3| x(p) E(p) ~= BQ(p). This situation evidently generalizes to
other rank 2 situations, such as Sp 4(Z) (discussed in x6) and G 2(Z). It is ho*
*ped
that the purely cohomological approach which we have presented here will help
yield some insight for understanding the cohomology of congruence subgroups,
but it remains a difficult undertaking.
16 ALEJANDRO ADEM
Detailed calculations for the cohomology of congruence subgroups with ratio-
nal coefficients can also be found in the paper by Lee and Schwermer [LS], we
refer the interested reader to this reference for further details. In their lan*
*guage,
the image of the cohomology of (p) in the cohomology of Q(p) corresponds to
the `cohomology at infinity' and the kernel can be identified with the `cuspidal
cohomology'.
References
[A] A. Ash, Cohomology of Congruence Subgroups of SLn(Z), Mathematische Anna*
*len 249
(1980), 55-73.
[AGG] A. Ash, Grayson, D., and Green, P., Computations of Cuspidal Cohomology *
*of Con-
gruence Subgroups of SL3(Z), J. Number Theory 19 (1984), 412-436.
[BS] A. Borel and Serre, J.-P., Corners and Arithmetic Groups, Comm. Math. He*
*lv. 48
(1973), 436-491.
[Br1] K. Brown, Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, *
*Springer-
Verlag, 1982.
[Br2] K. Brown, Buildings, Springer-Verlag, 1989.
[K] D. A. Kajdan, On the Connection of the Dual Space of a Group with the St*
*ructure of
its Closed Subgroups, Funct. Anal. and Appl. 1 (1967), 63-65.
[LS] R. Lee and Schwermer J., Cohomology of Arithmetic Subgroups of SL3(Z) at*
* Infinity,
J. Reine Angew. Math. 330 (1982), 100-131.
[S] J. Schwermer, On Arithmetic Quotients of the Siegel Upper Half Space of *
*Degree Two,
Compositio Math. 58 (1986), 233-258.
[Se] J.-P. Serre, Arithmetic Groups, Homological Group Theory, London Math. S*
*oc. (C.T.C.
Wall, ed.), vol. LNS 36, Cambridge University Press, 1979, pp. 105-136.
Mathematics Department, University of Wisconsin, Madison, WI 53706
E-mail address: adem@math.wisc.edu