Let Γ be a discrete group and p be a prime.
One of the fundamental results in group
cohomology is that H*(Γ, [ ]p)
is a finitely generated [ ]p-algebra if Γ
is a finite group
[8, 24]. The purpose of this paper is
to study the analogous question when Γ is no longer finite.Recall that Γ is said to have finite virtual cohomological dimension
(vcd) if there
exists a finite index torion-free subgroup Γ′ of Γ
such that Γ′ has finite cohomological
dimension over ℤ [4]. By definition vcd
Γ
is the cohomological dimension of Γ′. It is
easy to see that the mod p cohomology ring of a finite
vcd-group does not have to be
a finitely generated [ ]p-algebra in general.
For instance, if Γ is a countably infinite free product of ℤ's,
then
H1(Γ, [ ]p) is not
finite dimensional over [ ]p. The three most
important classes of examples of finite vcd-groups in which the
mod p cohomology ring is a finitely generated [ ]p-algebra
are arithmetic groups [2], mapping class groups
[9, 10] and outer automorphism groups of
free groups [5]. In each of these examples,
the proof of finite generation involves the construction of a specific
Γ-complex with
appropriate finiteness conditions. These constructions should be regarded
as utilizing
the geometry underlying these special classes of groups. In contrast, the
result we
prove will depend only on the algebraic structure of the group Γ.
Mapping class groups and outer automorphism groups of free groups are C∗-simple
