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

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.

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.

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     author = {Valette, Alain},
     title = {On the {Haagerup} inequality and groups acting on $\tilde{A}_n$-buildings},
     journal = {Annales de l'Institut Fourier},
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     year = {1997},
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Valette, Alain. On the Haagerup inequality and groups acting on $\tilde{A}_n$-buildings. Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 1195-1208. doi : 10.5802/aif.1596. https://aif.centre-mersenne.org/articles/10.5802/aif.1596/

[AO76] C.A Akemann and P.A. Ostrand, Computing norms in group C*-algebras, Amer. J. Math., 98 (1976), 1015-1047. | MR | Zbl

[BB] W. Ballmann and M. Brin, Orbihedra of nonpositive curvature, to appear in Invent. Math. | Numdam | Zbl

[BH78] A. Borel and G. Harder, Existence of discrete co-compact subgroups of reductive groups over local fields, J. für reine und angew. Math., 298 (1978), 53-64. | MR | Zbl

[CMSZ93] D. Cartwright, A. Mantero, T. Steger and A. Zappa, Groups acting simply transitively on the vertices of a building of type Ã2, Geometriae Dedicata, 47 (1993), 143-166. | MR | Zbl

[CoS93] D. Cartwright, W. Miotkowski and T. Steger, Property (T) and Ã2 groups, Ann. Inst. Fourier, Grenoble, 44-1 (1993), 213-248. | Numdam | Zbl

[CS] D. Cartwright, T. Steger, A family of Ãn groups, preprint. | Zbl

[CM90] A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29 (1990), 345-388. | Zbl

[FFR95] S.C. Ferry, A. Ranicki and J. Rosenberg (eds.), Novikov conjectures, index theorems and rigidity, London Math. Soc. Lect. Note Ser. 226, Cambridge U.P., 1995. | Zbl

[GW71] D. Gromoll and J.A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc., 77 (1971), 545-552. | MR | Zbl

[Haa79] U. Haagerup, An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math., 50 (1979), 279-293. | MR | Zbl

[dlH88] P. De La Harpe, Groupes hyperboliques, algèbres d'opérateurs, et un théorème de Jolissaint, C.R. Acad. Sci. Paris, Sér I, 307 (1988), 771-774. | Zbl

[dlHRV93] P. De La Harpe, A.G. Robertson and A. Valette, On the spectrum of the sum of generators of a finitely generated group, II, Colloquium Math., 65 (1993), 87-102. | MR | Zbl

[dlHV89] P. De La Harpe and A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175, Soc. Math. France, 1989.

[Jol90] P. Jolissaint, Rapidly decreasing functions in reduced C*-algebras of groups, Trans. amer. Math. Soc., 317 (1990), 167-196. | MR | Zbl

[Jol89] P. Jolissaint, K-theory of reduced C*-algebras and rapidly decreasing functions on groups, K-theory, 2 (1989), 723-735. | MR | Zbl

[Jol96] P. Jolissaint, An upper bound for the norms of powers of normalised adjacency operators, Pacific J. Math., 175, 432-436, 1996, Appendix to On spectra of simple random walks on one-relator groups, by P-A. Cherix and A. Valette. | Zbl

[JV91] P. Jolissaint and A. Valette, Normes de Sobolev et convoluteurs bornés sur L2(G), Ann. Inst. Fourier, Grenoble, 41-4 (1991), 797-822. | Numdam | Zbl

[Kes59] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354. | MR | Zbl

[Pan] P. Pansu, Formule de Matsushima, de Garland, et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles, Preprint Orsay, 1995.

[Pit] C. Pittet, Ends and isoperimetry, Preprint Neuchâtel, 1995.

[RRS] J. Ramagge, G. Robertson and T. Steger, A Haagerup inequality for Ã1 X Ã1 and Ã2 groups, Preprint, 1996. | Zbl

[Ron89] M. Ronan, Lectures on buildings, Academic Press, 1989. | MR | Zbl

[Swi] J. Swiatkowski, On the loop inequality for euclidean buildings, Ann. Inst. Fourier, Grenoble, 47-4 (1997), 1175-1194. | Numdam | MR | Zbl

[Tit72] J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. | MR | Zbl

[Tit86] J. Tits, Immeubles de type affine, in Buildings and the geometry of diagrams (L.A. Rosati, ed.), Lect. Notes in Math. 1181 (1986), Springer, 159-190. | MR | Zbl

[VSCC92] N. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and geometry on groups, Cambridge U.P., 1992. | MR | Zbl

[Zuk96] A. Zuk, La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres, C.R. Acad. Sci. Paris, 323 (1996), 453-458. | MR | Zbl

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