Group Theory/Topology
On a question of Serre
Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 741-744.

Consider an imaginary quadratic number field Q(m), with m a square-free positive integer, and its ring of integers O. The Bianchi groups are the groups SL2(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(m), où m est un entier positif ne contenant pas de carré, et son anneau dʼentiers O. Les groupes de Bianchi sont les groupes SL2(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.

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Published online:
DOI: 10.1016/j.crma.2012.09.001
Rahm, Alexander D. 1

1 National University of Ireland at Galway, Department of Mathematics, Ireland
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Rahm, Alexander D. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.09.001/

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