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

Author: Sam Hughes

Affiliation: University of Southampton

Abstract: We compute the equivariant $KO$-homology of the classifying
space for proper actions of $\textrm{SL}_3(\mathbb{Z})$ and 
$\textrm{GL}_3(\mathbb{Z})$. We also compute the Bredon homology and
equivariant $K$-homology of the classifying spaces for proper actions
of $\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])$ and 
$\textrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$ for each prime $p$. 
Finally, we prove the unstable Gromov-Lawson-Rosenberg conjecture 
for $\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])$ when 
$p\equiv11\pmod{12}$.