  We study completions of the group algebra of a finitely generated group and
relate nuclearity of such a completion to growth properties of the group. This
extends previous work of Jolissaint on nuclearity of rapidly decreasing
functions on a finitely generated group to more general weights than polynomial
decrease. The new group algebras and their duals are studied in detail and
compared to other approaches. As application we discuss the convergence of the
complete growth function introduced by Grigorchuk and Nagnibeda.
