On the equivariant $K$-homology of PSL${}_2$ of the imaginary quadratic integers

Rahm, Alexander D.

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.

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

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