Let be a discrete group. We give methods to compute, for a generalized (co)homology theory, its values on the Borel construction of a proper –CW–complex satisfying certain finiteness conditions. In particular we give formulas computing the topological –(co)homology and up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups these formulas are sharp. The main new tools we use for the –theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where is a finite group these theorems reduce to well-known results of Greenlees and Bökstedt.
Michael Joachim. Wolfgang Lück. "Topological $K$–(co)homology of classifying spaces of discrete groups." Algebr. Geom. Topol. 13 (1) 1 - 34, 2013. https://doi.org/10.2140/agt.2013.13.1