D.J. Benson
Complexity and varieties for infinite groups.
Preprint, 1996.

\begin{abstract}
This two part paper generalizes the usual notion of complexity and varieties
for modules over the group algebra of a finite group, to a
large class of infinite groups. The context is modules of type
$FP_\infty$ for groups in Kropholler's class \lhf. One of the
main results is that the category of such modules is
generated in a suitable sense by modules induced from finite
elementary abelian subgroups. This implies that an element of
complete cohomology of such a module is nilpotent if and only
if its restriction to every finite elementary abelian subgroup
is nilpotent. It also implies that the complexity
of a module of type $FP_\infty$ is finite, and that the variety
is supported on some finite collection of finite elementary 
abelian subgroups. An example is given which shows that the
complexity does not determine the rate of growth of the number
of generators in a projective resolution, in the way it does for
finite groups.
\end{abstract}