[Nombres de Betti des groupes localement compacts]
Cet article présente quelques résultats de la thèse de lʼauteur, « -Betti Numbers of Locally Compact Groups », dans laquelle la définition des nombres de Betti des groupes dénombrables a été généralisée au cas des groupes localement compacts unimodulaires. Dans de nombreux cas, les nombres 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 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, “-Betti Numbers of Locally Compact Groups”, wherein the definition of -Betti numbers for countable groups is extended to locally compact unimodular groups. In many cases, the -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.
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@article{CRMATH_2013__351_9-10_339_0,
author = {Henrik Densing Petersen},
title = {$ {L}^{2}${-Betti} numbers of locally compact groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {339--342},
publisher = {Elsevier},
volume = {351},
number = {9-10},
year = {2013},
doi = {10.1016/j.crma.2013.05.013},
language = {en},
}
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/
[1] Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. Math. Stud., vol. 94, Princeton University Press, Princeton, NJ, 1980
[2] Simplicity and superrigidity of twin building lattices, Invent. Math., Volume 176 (2009) no. 1, pp. 169-221 | DOI
[3] -cohomology and group cohomology, Topology, Volume 25 (1986) no. 2, pp. 189-215 | DOI
[4] Cohomology of buildings and their automorphism groups, Invent. Math., Volume 150 (2002) no. 3, pp. 579-627 | DOI
[5] Invariants de relations dʼéquivalence et de groupes, Publ. Math. Inst. Hautes Études Sci., Volume 95 (2002), pp. 93-150 | DOI
[6] Invariant percolation and harmonic Dirichlet functions, Geom. Funct. Anal., Volume 15 (2005) no. 5, pp. 1004-1051 | DOI
[7] Cohomologie des groupes topologiques et des algèbres de Lie, Textes Math., vol. 2, CEDIC, Paris, 1980
[8] -Betti numbers of locally compact groups and their cross section equivalence relations (preprint) | arXiv
[9] Dimension theory of arbitrary modules over finite von Neumann algebras and -Betti numbers. I. Foundations, J. Reine Angew. Math., Volume 495 (1998), pp. 135-162
[10] Dimension theory of arbitrary modules over finite von Neumann algebras and -Betti numbers. II. Applications to Grothendieck groups, -Euler characteristics and Burnside groups, J. Reine Angew. Math., Volume 496 (1998), pp. 213-236
[11] -Invariants: Theory and Applications to Geometry and K-Theory, Ergeb. Math. Grenzgeb. (3), vol. 44, Springer-Verlag, Berlin, 2002
[12] Continuous Bounded Cohomology of Locally Compact Groups, Lect. Notes Math., vol. 1758, Springer-Verlag, Berlin, 2001
[13] -Betti numbers of locally compact groups, University of Copenhagen, 2013 (Thesis) | arXiv
[14] Group cocycles and the ring of affiliated operators, Invent. Math., Volume 185 (2011) no. 3, pp. 561-592 | DOI
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