Complexity and Varieties for Infinite Groups, I

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 typeFP_∞ for groups in Kropholler's classLH$\text{F}$. 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 typeFP_∞ 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.