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

Hughes, Sam

doi:10.2140/agt.2021.21.3483

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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

DOI: 10.2140/agt.2021.21.3483

Abstract

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

Keywords

Baum–Connes conjecture, Gromov–Lawson–Rosenberg conjecture, special linear groups

Mathematical Subject Classification

Primary: 19L47, 53C21

Secondary: 19K99, 20J99, 55N91, 57R15

References

Publication

Received: 20 April 2020

Revised: 31 December 2020

Accepted: 9 January 2021

Published: 28 December 2021

Authors

Sam Hughes
School of Mathematical Sciences University of Southampton Southampton United Kingdom

Volume 21, issue 7 (2021)