On a question of Serre

Alexander D. Rahm

doi:10.1016/j.crma.2012.09.001

Group Theory/Topology

On a question of Serre

Presented by: Jean-Pierre Serre

Alexander D. Rahm ¹

¹ National University of Ireland at Galway, Department of Mathematics, Ireland

Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 741-744.

is referenced by Corrigendum to “On a question of Serre” [C. R. Acad. Sci. Paris, Ser. I 350 (2012) 741–744]

Abstract
Résumé

Consider an imaginary quadratic number field $Q (\sqrt{- m})$ , with m a square-free positive integer, and its ring of integers $O$ . The Bianchi groups are the groups $S L_{2} (O)$ . Further consider the Borel–Serre compactification [7] (1970) of the quotient of hyperbolic 3-space $H$ by a finite index subgroup Γ in a Bianchi group, and in particular the following question which Serre posed on p. 514 of the quoted article. Consider the map α induced on homology when attaching the boundary into the Borel–Serre compactification. How can one determine the kernel of α (in degree 1)? of the kernel of α. In the quoted article, Serre did add the question what submodule precisely this kernel is. Through a local topological study, we can decompose the kernel of α into its parts associated to each cusp.

Considérons un corps quadratique imaginaire $Q (\sqrt{- m})$ , où m est un entier positif ne contenant pas de carré, et son anneau dʼentiers $O$ . Les groupes de Bianchi sont les groupes $S L_{2} (O)$ . Puis, nous considérons la compactification de Borel–Serre [7] (1970) du quotient de lʼespace hyperbolique $H$ à trois dimensions par un sous-groupe Γ dʼindice fini dans un groupe de Bianchi, et en particulier la question suivante que Serre posait sur la p. 514 de lʼarticle cité. Considérons lʼapplication α induite en homologie quand le bord est attaché dans la compactification de Borel–Serre. Comment peut-on déterminer le noyau de α (en degré 1) ? Serre se servait dʼun argument topologique global et obtenait le rang du noyau de α. Dans lʼarticle cité, Serre rajoutait la question de quel sous-module précisément il sʼagit pour ce noyau. A travers dʼune étude topologique locale, nous pouvons décomposer le noyau de α dans ses parties associées à chacune des pointes.

Received: 2012-07-27
Accepted: 2012-09-04
Published online: 2012-08-01

DOI: 10.1016/j.crma.2012.09.001

Author's affiliations:

Alexander D. Rahm ¹

¹ National University of Ireland at Galway, Department of Mathematics, Ireland

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Alexander D. Rahm. On a question of Serre. Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 741-744. doi : 10.1016/j.crma.2012.09.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.001/

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