On the equivariant $K$- and $KO$-homology of some special linear groups

We compute the equivariant $KO$-homology of the classifying space for proper actions of ${\text{SL}}_3(\mathbb{Z})$ and ${\text{GL}}_3(\mathbb{Z})$. We also compute the Bredon homology and equivariant $K$-homology of the classifying spaces for proper actions of ${\text{PSL}}_2(\mathbb{Z}[\frac{1}{p}])$ and ${\text{SL}}_2(\mathbb{Z}[\frac{1}{p}])$ for each prime $p$. Finally, we prove the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.