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

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 $\textrm{SL}_3(\mathbb{Z})$ 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 $p\equiv11\pmod{12}$.

math.KT, Equivariant K-theory, Gromov-Lawson-Rosenberg conjecture, Baum-Connes conjecture

Hughes, Sam

a41196d7-14a9-42f8-b6c1-95e00f98910a

Hughes, Sam

a41196d7-14a9-42f8-b6c1-95e00f98910a

Hughes, Sam
(2021)
On the equivariant K- and KO-homology of some special linear groups.
*Algebraic & Geometric Topology*.
(In Press)

## 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 $p\equiv11\pmod{12}$.

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** 2004.08199v1
- Author's Original**
Text

** On the equivariant K- and KO-homology of some special linear groups
- Accepted Manuscript**
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## More information

Submitted date: 17 April 2020

Accepted/In Press date: 10 January 2021

Keywords:
math.KT, Equivariant K-theory, Gromov-Lawson-Rosenberg conjecture, Baum-Connes conjecture

## Identifiers

Local EPrints ID: 439571

URI: http://eprints.soton.ac.uk/id/eprint/439571

ISSN: 1472-2747

PURE UUID: 995de742-d646-4b19-8836-30746b33cbbc

## Catalogue record

Date deposited: 27 Apr 2020 16:31

Last modified: 16 Mar 2024 07:36

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## Contributors

Author:
Sam Hughes
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