Homology and <i>K</i>-theory of the Bianchi groups

Alexander D. Rahm

doi:10.1016/j.crma.2011.05.014

Homological Algebra/Group Theory

Homology and K-theory of the Bianchi groups
[Homologie et K-théorie des groupes de Bianchi]

Présenté par : Christophe Soulé

Alexander D. Rahm ¹

¹ Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 615-619.

Résumé
Abstract

Nous mettons en évidence une correspondance entre la torsion homologique des groupes de Bianchi et de nouveaux invariants géométriques, calculables grâce à leur action sur lʼespace hyperbolique. Nous lʼutilisons pour calculer explicitement leur homologie de groupe à coefficients entiers et leur K-homologie équivariante. En conséquence de la conjecture de Baum/Connes, qui est vérifiée pour ces groupes, nous obtenons la K-théorie de leurs C*-algèbres réduites en termes dʼimages isomorphes de la K-homologie calculée. Nous trouvons dʼailleurs une application à la cohomologie dʼorbi-espace de Chen/Ruan.

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant K-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the K-theory of their reduced $C^{⁎}$ -algebras in terms of isomorphic images of the computed K-homology. We further find an application to Chen/Ruan orbifold cohomology.

Reçu le : 2011-03-04
Accepté le : 2011-05-30
Publié le : 2011-06-01

DOI : 10.1016/j.crma.2011.05.014

Affiliations des auteurs :

Alexander D. Rahm ¹

¹ Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

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Alexander D. Rahm. Homology and K-theory of the Bianchi groups. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 615-619. doi : 10.1016/j.crma.2011.05.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.05.014/

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