COMPLEXITY AND VARIETIES FOR INFINITE GROUPS .1.

This two-part paper generalizes the usual notion of complexity and var ieties for modules over the group algebra of a finite group, to a larg e class of infinite groups. The context is modules of type FPinfinity for groups in Krophaller's class LHF. One of the main results is that the category of such modules is generated in a suitable sense by modul es induced from finite elementary abelian subgroups. This implies that an element of complete cohomology of such a module is nilpotent if an d only if its restriction to every finite elementary abelian subgroup is nilpotent. It also implies that the complexity of a module of type FPchi is finite, and that the variety is supported on some finite coll ection of finite elementary abelian subgroups. An example is given whi ch shows that the complexity does not determine the rate of growth of the number of generators in a projective resolution, in the way it doe s for finite groups. (C) 1997 Academic Press.