  We define a notion of complexity for modules over infinite groups. We show
that if $M$ is a module over the group ring $kG$, and $M$ has complexity $\leq
f$ (where $f$ is some complexity function) over some set of finite index
subgroups of $G$, then $M$ has complexity $\leq f$ over $G$ (up to a direct
summand). This generalizes the Alperin-Evens Theorem, which states that if the
group $G$ is finite then the complexity of $M$ over $G$ is the maximal
complexity of $M$ over an elementary abelian subgroup of $G$. We also show how
we can use this generalization in order to construct projective resolutions for
the integral special linear groups, $SL(n,\Z)$, where $n\geq 2$.
