On the equivariant K-homology of PSL 2 of the imaginary quadratic integers
[Sur la K-homologie équivariante de PSL 2 sur les entiers quadratiques imaginaires]
Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1667-1689.

Pour la K-homologie équivariante de tous les groupes de Bianchi (PSL 2 sur les entiers quadratiques imaginaires), nous démontrons des formules pour la partie due à la torsion, en termes de quantités élémentaires de la théorie des nombres. Pour arriver à cette fin, nous introduisons une nouvelle technique pour le calcul de l’homologie de Bredon : un scindage des anneaux de représentation, qui nous permet d’adapter la technique récente de réduction des sous-complexes de torsion, développée pour l’homologie des groupes, à notre usage pour l’homologie de Bredon.

We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL 2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.

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DOI : 10.5802/aif.3047
Classification : 55N91, 19L47
Keywords: Equivariant homology and cohomology, Equivariant $K$-theory, Bianchi groups, PSL$_2$ of the imaginary quadratic integers
Mot clés : Homologie et cohomologie équivariantes, $K$-théorie équivariante, Groupes de Bianchi, PSL$_2$ sur les entiers quadratiques imaginaires
Rahm, Alexander D. 1

1 Université du Luxembourg, Mathematics Research Unit 6, rue Coudenhove-Kalergi L-1359 Luxembourg-Kirchberg Luxembourg
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Rahm, Alexander D. On the equivariant $K$-homology  of PSL$_2$ of the imaginary quadratic integers. Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1667-1689. doi : 10.5802/aif.3047. http://www.numdam.org/articles/10.5802/aif.3047/

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