2000]Primary: 43A15; Secondary: 20F65, 20F18 GROUP COHOMOLOGY AND L-COHOMOLOGY OF FINITELY GENERATED GROUPS

In this paper G will always be a finitely generated, infinite group and S will always be a symmetric generating set for G. Let M be a right G-module. A 1cocycle with values in M is a map δ : G → M such that δ(gh) = (δ(h)) g + δ(g) for any g, h ∈ G; a 1-coboundary is a 1-cocycle of the form δ(g) = xg − x for some x ∈ M and for all g ∈ G. We denote by Z (G,M) the vector space of all 1-cocycles and the vector space of all 1-coboundaries will be denoted by B (G,M) . The factor groupH (G,M) = Z (G,M) /B (G,M) is called the first cohomology group of G with coefficients in M . Suppose now that M is a topological vector space and that the action of G on M is continuous. Then we give Z (G,M) the compact open topology. Assuming that M is Hausdorff, this means that δn → δ in Z 1 (G,M) if and only if δn(g) → δ(g) in M for all g ∈ G. In general B 1 (G,M) is not closed in Z (G,M). The quotient space H 1 (G) = Z (G,M) /B1 (G,M), where B1 (G,M) is the closure of B (G,M) in Z (G,M), is called the first reduced cohomology space. Let F (G) be the set of complex-valued functions on G. We may represent each f in F (G) as a formal sum ∑