#### Volume 21, issue 7 (2021)

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

### Sam Hughes

Algebraic & Geometric Topology 21 (2021) 3483–3512
##### Abstract

We compute the equivariant $K\phantom{\rule{-0.17em}{0ex}}O$–homology of the classifying space for proper actions of ${SL}_{3}\left(ℤ\right)$ and ${GL}_{3}\left(ℤ\right)$. We also compute the Bredon homology and equivariant $K$–homology of the classifying spaces for proper actions of ${PSL}_{2}\left(ℤ\left[\frac{1}{p}\right]\right)$ and ${SL}_{2}\left(ℤ\left[\frac{1}{p}\right]\right)$ 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.

##### Keywords
Baum–Connes conjecture, Gromov–Lawson–Rosenberg conjecture, special linear groups
##### Mathematical Subject Classification
Primary: 19L47, 53C21
Secondary: 19K99, 20J99, 55N91, 57R15