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On the Haagerup inequality and groups acting on \(\tilde A_ n\)-buildings. (English) Zbl 0886.51003
Summary: Let \(\Gamma\) be a group endowed with a length function \(L\), and let \(E\) be a linear subspace of \(\mathbb{C}\Gamma\). We say that \(E\) satisfies the Haagerup inequality if there exist constants \(C,s>0\) such that, for any \(f\in E\), the convolutor norm of \(f\) on \(\ell^{2}(\Gamma)\) is dominated by \(C\) times the \(\ell^{2}\) norm of \(f(1+L)^{s}\). We show that, for \(E=\mathbb{C}\Gamma\), 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 \(\text{Rad}_{L}(\Gamma)\) 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 \(\text{Rad}_{L}(\Gamma)\) 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 \(\tilde{A}_{n}\), the space \(\text{Rad}_{L}(\Gamma)\) satisfies the Haagerup inequality, and \(\Gamma\) is non-amenable.

MSC:
51E24 Buildings and the geometry of diagrams
60G50 Sums of independent random variables; random walks
44A35 Convolution as an integral transform
43A05 Measures on groups and semigroups, etc.
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References:
[1] C.A AKEMANN and P.A. OSTRAND, Computing norms in group C*-algebras, Amer. J. Math., 98 (1976), 1015-1047. · Zbl 0342.22008
[2] W. BALLMANN and M. BRIN, Orbihedra of nonpositive curvature, to appear in Invent. Math. · Zbl 0866.53029
[3] 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. · Zbl 0385.14014
[4] 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. · Zbl 0784.51011
[5] D. CARTWRIGHT, W. MIOTKOWSKI and T. STEGER, Property (T) and ã2 groups, Ann. Inst. Fourier, Grenoble, 44-1 (1993), 213-248. · Zbl 0792.43002
[6] D. CARTWRIGHT, T. STEGER, A family of ãn groups, preprint. · Zbl 0923.51010
[7] A. CONNES and H. MOSCOVICI, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29 (1990), 345-388. · Zbl 0759.58047
[8] 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 0954.57018
[9] 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. · Zbl 0237.53037
[10] U. HAAGERUP, An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math., 50 (1979), 279-293. · Zbl 0408.46046
[11] 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 0653.46059
[12] 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. · Zbl 0846.46036
[13] P. de la HARPE and A. VALETTE, La propriété (T) de Kazhdan pour LES groupes localement compacts, Astérisque 175, Soc. Math. France, 1989. · Zbl 0759.22001
[14] P. JOLISSAINT, Rapidly decreasing functions in reduced C*-algebras of groups, Trans. amer. Math. Soc., 317 (1990), 167-196. · Zbl 0711.46054
[15] P. JOLISSAINT, K-theory of reduced C*-algebras and rapidly decreasing functions on groups, K-theory, 2 (1989), 723-735. · Zbl 0692.46062
[16] 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 0865.60059
[17] P. JOLISSAINT and A. VALETTE, Normes de Sobolev et convoluteurs bornés sur L2(G), Ann. Inst. Fourier, Grenoble, 41-4 (1991), 797-822. · Zbl 0734.43002
[18] H. KESTEN, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354. · Zbl 0092.33503
[19] P. PANSU, Formule de Matsushima, de Garland, et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles, Preprint Orsay, 1995.
[20] C. PITTET, Ends and isoperimetry, Preprint Neuchâtel, 1995.
[21] J. RAMAGGE, G. ROBERTSON and T. STEGER, A Haagerup inequality for ã1 X ã1 and ã2 groups, Preprint, 1996. · Zbl 0906.43009
[22] M. RONAN, Lectures on buildings, Academic Press, 1989. · Zbl 0694.51001
[23] J. SWIATKOWSKI, On the loop inequality for Euclidean buildings, Ann. Inst. Fourier, Grenoble, 47-4 (1997), 1175-1194. · Zbl 0886.51004
[24] J. TITS, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. · Zbl 0236.20032
[25] 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. · Zbl 0611.20026
[26] N. VAROPOULOS, L. SALOFF-COSTE and T. COULHON, Analysis and geometry on groups, Cambridge U.P., 1992. · Zbl 0813.22003
[27] A. ZUK, La propriété (T) de Kazhdan pour LES groupes agissant sur LES polyèdres, C.R. Acad. Sci. Paris, 323 (1996), 453-458. · Zbl 0858.22007
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