On the Haagerup inequality and groups acting on A ˜ n -buildings
Annales de l'Institut Fourier, Volume 47 (1997) no. 4, p. 1195-1208

Let Γ be a group endowed with a length function L, and let E be a linear subspace of CΓ. We say that E satisfies the Haagerup inequality if there exists constants C,s>0 such that, for any fE, the convolutor norm of f on 2 (Γ) is dominated by C times the 2 norm of f(1+L) s . We show that, for E=CΓ, the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on Γ. If L is a word length function on a finitely generated group Γ, we show that, if the space Rad L (Γ) of radial functions with respect to L satisfies the Haagerup inequality, then Γ is non-amenable if and only if Γ has superpolynomial growth. We also show that the Haagerup inequality for Rad L (Γ) has a purely combinatorial interpretation; thus, using the main result of the companion paper by J. Swiatkowski, we deduce that, for a group Γ acting simply transitively on the vertices of a thick euclidean building of type A ˜ n , the space Rad L (Γ) satisfies the Haagerup inequality, and Γ is non-amenable.

Soit Γ un groupe muni d’une fonction-longueur L, et soit E un sous-espace vectoriel de CΓ. On dira que E satisfait à l’inégalité de Haagerup s’il existe des constantes C,s>0 telles que, pour tout fE, la norme de convolution de f sur 2 (Γ) soit dominée par C fois la norme 2 de f(1+L) s . Nous montrons que, pour E=CΓ, l’inégalité de Haagerup s’exprime en termes de décroissance des marches aléatoires associées à des mesures de probabilité symétriques à support fini sur Γ. Si L est la longueur des mots sur un groupe Γ de type fini, nous montrons que, si l’espace Rad L (Γ) des fonctions radiales par rapport à L satisfait à l’inégalité de Haagerup, alors Γ est non moyennable si et seulement si Γ est à croissance superexponentielle. Nous montrons aussi que l’inégalité de Haagerup pour Rad L (Γ) a une interprétation purement combinatoire; en utilisant le résultat principal de l’article de J. Swiatkowski dans le même fascicule, nous déduisons que, pour un groupe Γ opérant simplement transitivement sur les sommets d’un immeuble euclidien épais de type A ˜ n , l’espace Rad L (Γ) satisfait à l’inégalité de Haagerup, et Γ est non moyennable.

@article{AIF_1997__47_4_1195_0,
     author = {Valette, Alain},
     title = {On the Haagerup inequality and groups acting on $\tilde{A}\_n$-buildings},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {4},
     year = {1997},
     pages = {1195-1208},
     doi = {10.5802/aif.1596},
     zbl = {0886.51003},
     mrnumber = {99f:43001},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1997__47_4_1195_0}
}
Valette, Alain. On the Haagerup inequality and groups acting on $\tilde{A}_n$-buildings. Annales de l'Institut Fourier, Volume 47 (1997) no. 4, pp. 1195-1208. doi : 10.5802/aif.1596. http://www.numdam.org/item/AIF_1997__47_4_1195_0/

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