Equivariant K-homology and K-theory for some discrete planar affine groups

Abstract

We consider the semi-direct products $G={\mathbb{Z}}^2⋊G{L}_2(\mathbb{Z}),{\mathbb{Z}}^2⋊S{L}_2(\mathbb{Z})$ and ${\mathbb{Z}}^2⋊\Gamma (2)$ (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 $C^{\ast }$-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 second and third groups, the computations in $K_0$ provide explicit generators that are matched by the Baum-Connes assembly map.

v2: minor changes, to appear in IMRN, accepted version