Cohomology of discrete groups and their finite subgroups. (English) Zbl 0982.55010

The author starts from the result that if \(G\) is a finite group, and the Abelian subgroups are suitably ordered then \(H^*(G,\mathbb{Z})\cong \varprojlim_A H^*(A, \mathbb{Z})\) modulo a nilpotent kernel and cokernel. By means of a Leray spectral sequence argument he extends this result to discrete groups of finite virtual cohomological dimension, and replaces the coefficients \(\mathbb{Z}\) by an arbitrary commutative ring \(k\) with unit. He also shows that it suffices to take the limit over a category \({\mathcal J}\) of finite subgroups which satisfy the conditions (i) for every finite subgroup \(H\) there exists \(K\) in \({\mathcal J}\) with \(H\subseteq K\) and (ii) \({\mathcal J}\) is closed under conjugation and taking intersections. There is an analogous result for Tate-Farrell cohomology and (presumably) for the more general cohomology theory applying to a wider class of discrete groups.


55R40 Homology of classifying spaces and characteristic classes in algebraic topology
20J06 Cohomology of groups
20J05 Homological methods in group theory
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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