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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, wecompute both sides of the Baum-Connes conjecture, namely the equivariant$K$-homology of the classifying space $\underline{E}G$ for proper actions onthe 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 madepossible by the existence of a 3-dimensional model for $\underline{E}G$, whichallows to replace equivariant K-homology by Bredon homology. We pay dueattention to the presence of torsion in $G$, leading to an extensive study ofthe wallpaper groups associated with finite subgroups. For the second and thirdgroups, the computations in $K_0$ provide explicit generators that are matchedby the Baum-Connes assembly map.

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