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Valette, Alain

Nom
Valette, Alain
Affiliation principale
Site web
Fonction
Professeur.e ordinaire
Email
alain.valette@unine.ch
Identifiants
https://libra.unine.ch/handle/123456789/455
_
0000-0002-2237-8384

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  • Publication
    Métadonnées seulement
    Isometric group actions on hilbert spaces: Growth of cocycles
    (2007)
    De Cornulier, Yves
    ;
    Tessera, Romain
    ;
    Valette, Alain 
    We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into a Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled" Folner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.
  • Publication
    Métadonnées seulement
  • Publication
    Métadonnées seulement
    Group cohomology, harmonic functions and the first L
    (1997)
    Bekka, Bachir
    ;
    Valette, Alain 
    For an infinite, finitely generated group Gamma, we study the first cohomology group H-1(Gamma, lambda(Gamma)) with coefficients in the left regular representation lambda(Gamma) of Gamma on l(2)(Gamma). We first prove that H-1(Gamma, C Gamma) embeds into H-1(Gamma, lambda(Gamma)); as a consequence, if H-1(Gamma, lambda(Gamma)) = 0, then Gamma is not amenable with one end. For a Cayley graph X of Gamma, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Gamma is not amenable, then there is a natural isomorphism between H-1(Gamma, lambda(Gamma)) and HD(X)/C (the latter space being isomorphic to the first L-2-cohomology space of Gamma). We draw the following consequences: (1) If Gamma has infinitely many ends, then HD(X) not equal C; (2) If Gamma has Kazhdan's property (T), then HD(X) = C; (3) The property H-1(Gamma, lambda(Gamma)) = 0 is a quasi-isometry invariant; (4) Either H-1(Gamma, lambda(Gamma)) = 0 or H-1(Gamma, lambda(Gamma)) is infinite-dimensional; (5) If Gamma = Gamma(1) x Gamma(2) with Gamma(1) non-amenable and Gamma(2) infinite, then H-1(Gamma, lambda(Gamma)) = 0.