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

Abstract

Abstract 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.