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).



Which $K$-groups $K(C^*_r(G))$ are computed?
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)](http://aif.cedram.org/aif-bin/fitem?id=AIF_2016__66_4_1667_0)


Computations for Heisenberg-type groups have been established in the thesis of Olivier Isely [(link)](https://doc.rero.ch/record/28552/files/Isely_Olivier_-_K-theory_and_K-homology_for_semi-direct_products_of_Z2_by_Z_UNINE_THESE_2246_2011.pdf)


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)](https://arxiv.org/abs/math/0604402)


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)](https://arxiv.org/abs/1707.05133)


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](http://www.him.uni-bonn.de/lueck/).


