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

Henrik Densing Petersen

doi:10.1016/j.crma.2013.05.013

Group Theory

L^{2}

-Betti numbers of locally compact groups

Presented by: the Editorial Board

Henrik Densing Petersen ¹

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

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

Abstract
Résumé

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.

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.

Received: 2013-03-20
Accepted: 2013-05-28
Published online: 2013-05-01

DOI: 10.1016/j.crma.2013.05.013

Author's affiliations:

Henrik Densing Petersen ¹

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

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     title = {$ {L}^{2}${-Betti} numbers of locally compact groups},
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Henrik Densing Petersen. $ {L}^{2}$-Betti numbers of locally compact groups. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 339-342. doi : 10.1016/j.crma.2013.05.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.05.013/

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