$ {L}^{2}$-Betti numbers of locally compact groups

Petersen, Henrik Densing

doi:10.1016/j.crma.2013.05.013

Group Theory

L^{2}

-Betti numbers of locally compact groups
[Nombres

L^{2}

de Betti des groupes localement compacts]

Présenté par : the Editorial Board

Petersen, Henrik Densing ¹

¹ SB-MATHGEOM-EGG, EPFL, Station 8, CH-1015, Lausanne, Switzerland

Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 339-342.

Résumé
Abstract

Cet article présente quelques résultats de la thèse de lʼauteur, « $L^{2}$ -Betti Numbers of Locally Compact Groups », dans laquelle la définition des nombres $L^{2}$ de Betti des groupes dénombrables a été généralisée au cas des groupes localement compacts unimodulaires. Dans de nombreux cas, les nombres $L^{2}$ de Betti dʼun réseau quelconque dans un groupe localement compact sont proportionnels à ceux du groupe localement compact ambiant. On peut donc utiliser des théories puissantes concernant certaines classes de groupes localement compacts pour obtenir des calculs des nombres $L^{2}$ de Betti de leurs réseaux, en particulier les réseaux arithmétiques et les réseaux de Kac–Moody.

The present paper is a summary and overview of results obtained in the authorʼs thesis, “ $L^{2}$ -Betti Numbers of Locally Compact Groups”, wherein the definition of $L^{2}$ -Betti numbers for countable groups is extended to locally compact unimodular groups. In many cases, the $L^{2}$ -Betti numbers of a lattice are proportional to those of the ambient locally compact group, yielding new results for certain classes of lattices, including arithmetic lattices and Kac–Moody lattices.

Reçu le : 2013-03-20
Accepté le : 2013-05-28
Publié le : 2013-05-01

DOI : 10.1016/j.crma.2013.05.013

Affiliations des auteurs :

Petersen, Henrik Densing ¹

¹ SB-MATHGEOM-EGG, EPFL, Station 8, CH-1015, Lausanne, Switzerland

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     pages = {339--342},
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Petersen, Henrik Densing. $ {L}^{2}$-Betti numbers of locally compact groups. Comptes Rendus. Mathématique, Tome 351 (2013) no. 9-10, pp. 339-342. doi : 10.1016/j.crma.2013.05.013. http://www.numdam.org/articles/10.1016/j.crma.2013.05.013/

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