Equivariant K-Homology and K-Theory for Some Discrete Planar Affine Groups.

We consider the semi-direct products |$G={\mathbb{Z}}^{2}\rtimes GL_{2}({\mathbb{Z}}), {\mathbb{Z}}^{2}\rtimes SL_{2}({\mathbb{Z}})$|⁠ , and |${\mathbb{Z}}^{2}\rtimes \Gamma (2)$| (where |$\Gamma (2)$| is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum–Connes conjecture, namely the equivariant |$K$| -homology of the classifying space |$\underline{E}G$| for proper actions on the left-hand side, and the analytical K-theory of the reduced group |$C^{*}$| -algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for |$\underline{E}G$|⁠ , which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in |$G$|⁠ , leading to an extensive study of the wallpaper groups associated with finite subgroups. For the first and third groups, the computations in |$K_{0}$| provide explicit generators that are matched by the Baum–Connes assembly map. [ABSTRACT FROM AUTHOR]