On the \(K\)-theory of the classifying space of a discrete group
The author establishes a decomposition for the equivariant \(K\)-theory of a discrete group \(\Gamma\) with suitable assumptions. Let \(X\) be an admissible \(\Gamma\)-complex (i.e.: a finite-dimensional contractible \(\Gamma\)-complex such that the finite subgroups are exactly the isotropy subgroups and the fixed-point set \(X^ H\) are contractible) and let \(\Gamma'\) be a normal torsion-free subgroup in \(\Gamma\) with finite factor group \(G\), where \(\Gamma\) have finite virtual cohomological dimension. Then \(\Gamma'\) acts freely on \(X\) and we have, by the completion theorem, an isomorphism: \(K^*_ G(X/\Gamma')^ \wedge\cong K^*(B\Gamma)\). Using a recent result of Atiyah and Segal (expressing \(K^*_ G(X)\otimes\mathbb{C}\) as a sum of \(K^*(Y^{\langle g\rangle}/C(g))\otimes\mathbb{C}\), as \(g\) ranges over conjugacy classes of elements in \(G)\), the author indicates that the \(K\)-theory of \(B\Gamma\) can be calculated given enough information about its elements of finite order and their centralizers --- see formula (3.1). Specific examples presented at the end of this note include amalgamated products as well as certain arithmetic groups.
scientific article; zbMATH DE number 56284