Which $K$-groups $K(C^*_r(G))$ are computed?

Question

We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map for possible computation of the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ for a topological, locally compact, second-countable Hausdorff group $G$.

Up to now I have not seen much computations of $K(C^*_r(G))$.

Has anyone references to such computations, in particular in computing the left hand side of the Baum-Connes map, under the Chern map. That is, computations of the Czech cohomology groups $$\lim_{X \subseteq \underline EG} H(X,G)$$ (something like that).

Answer 1

Here are some known computations for infinite discrete groups. Basically, most of these proceed by computing the equivariant K-homology of the classifying space of proper actions and deduce the computation for K-theory of the group $C^\ast$-algebra via the assembly map.

For the Bianchi groups:

• A.D. Rahm. On the equivariant K-homology of ${\rm PSL}_2$ of the imaginary quadratic integers. Ann. Inst. Fourier 66 (2016), 1667-1689. (link to journal page)

Computations for Heisenberg-type groups have been established in the thesis of Olivier Isely (link)

Right-angled Coxeter groups:

• R. Sanchez-Garcia: Equivariant K-homology for some Coxeter groups. J. London Math. Soc. 75 (2007), 773-790. (link to arXiv)

For hyperbolic reflection groups:

• J-F. Lafont, I.J. Ortiz, A.D. Rahm, R.J. Sanchez-Garcia: Equivariant K-homology for hyperbolic reflection groups. arXiv:1707.05133 (link to arXiv)

The last paper also contains discussion and many further literature references to further computations of K-theory of group $C^\ast$-algebras, most notably by Wolfgang Lück and collaborators. There is also a book in progress on the isomorphism conjectures which contains a chapter on computations, see Wolfgang Lück's homepage.

Answer 2

There are some recent papers of Valette and coauthors which give explicit calculations/descriptions of the K-theory for some of the Baumslag-Solitar groups, namely BS($1,n$) (Pooya and Valette, arXiv 1604.05607), and also for certain lamplighters over ${\bf Z}$ (Flores, Pooya and Valette, arXiv 1610.02798). In both cases, the authors determine the LHS and the RHS of the Baum-Connes "picture" separately, and then verify explicitly that the BC map is an isomorphism.

(Apologies if these examples are covered in the references already provided by Matthias Wendt.)