Contemporary Mathematics
Volume 220, 1998
Buildings, Group Extensions and the
Cohomology of Congruence Subgroups
Alejandro Adem
Introduction
Let Tn denote the Tits Building associated to the discrete group SLn(Z). In this
note we will be interested in computing the equivariant cohomology Hr(ITnJ, Z),
where JTnl is the geometric realization of Tn, and
r
is a torsion-free discrete sub-
group of finite index in SLn(Z). In the special case n
=
3, we will see that this
equivariant cohomology in fact corresponds to the ordinary cohomology of a
4-
dimensional Poincare Duality group which has
r
as its quotient.
This is then applied to partially compute the cohomology of the level p con-
gruence subgroups (p an odd prime) in SL3 (Z). In particular we obtain
THEOREM.
'Let r(p) denote the level p congruence subgroup of SL3(Z); then
di111Q H
3(r(p),
Q)
;?:
12
1
(p3
-
1)(p3
-
3p2
- P
+
15)
+
1.
It turns out that this method can be used for other rank two groups such as
Sp4(Z) and G2(Z). In §6 we apply this to the level
p
congruence subgroups in
Sp4(Z), obtaining the cohomology of the relevant parabolic subgroups and from
this a lower bound for the fourth betti number:
THEOREM.
Let r(p) denote the level p congruence subgroup in Sp4(Z); then
1
di111Q H
4
(r(p), Q)
;?:
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 congruence
subgroups are homomorphic images of Poincare duality groups with kernel a free,
infinitely generated group. These groups are interesting in their own right, and
probably deserve further attention. They can be realized as the fundamental groups
of the Borel-Serre compactification; this is discussed in §7.
1991 Mathematics Subject Classification. Primary 20J Secondary 55R.
Partially supported by an NSF Grant
©
1998 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/220/03090
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