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Valette, Alain

Nom
Valette, Alain
Affiliation principale
Site web
Fonction
Professeur ordinaire
Email
alain.valette@unine.ch
Identifiants
https://libra.unine.ch/handle/123456789/455

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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
    On the Haagerup inequality and groups acting on Ã
    (1997)
    Valette, Alain 
    Let Gamma be a group endowed with a length function L, and let E be a linear subspace of C Gamma. We say that E satisfies the Haagerup inequality if there exists constants C, s > 0 such that, for any f is an element of E, the convolutor norm of f on l(2)(Gamma) is dominated by C times the l(2) norm of f(l + L)(s). We show that, for E = C Gamma, the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on Gamma. If L is a word length function on a finitely generated group Gamma, we show that, if the space Rad(L)(Gamma) of radial functions with respect to L satisfies the Haagerup inequality, then Gamma is non-amenable if and only if Gamma has superpolynomial growth. We also show that the Haagerup inequality for Rad(L)(Gamma) has a purely combinatorial interpretation; thus, using the main result of the companion paper by J. Swiatkowski, we deduce that, for a group Gamma acting simply transitively on the vertices of a thick euclidean building of type (A) over tilde(n), the space Rad(L)(Gamma) satisfies the Haagerup inequality, and Gamma is non-amenable.