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 SL3(Z) and GL3(Z). We also compute the Bredon homology and equivariant K–homology of the classifying spaces for proper actions of PSL2(Z[1p]) and SL2(Z[1p]) 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.