Explicit Kazhdan constants for representations of semisimple and arithmetic groups

Annales de l'Institut Fourier

Volumn 50, Issue 3, 2000, Pages 833-863

  Shalom, Yehuda ^a  

^a YALE UNIVERSITY   (United States)

Author keywords

Arithmetic groups; Kazhdan constants; Lattices; Property (T); Semisimple groups

Indexed keywords

EID: 0034412459     PISSN: 03730956     EISSN: None     Source Type: Journal    
DOI: 10.5802/aif.1775     Document Type: Article

References (59)

1. M.W. BALDONI SILVA, Unitary dual of Sp(n, 1), n ≥ 2, Duke Math. Journal, 48 (1981), 549-583
2. 2-cohomology and property (T) for automorphism groups of polyhedral cell complexes, GAFA, 7 (1997), 615-645.
3. M.W. BALDONI SILVA and D. BARBASCH, The unitary spectrum for real rank one groups, Invent. Math., 72 (1983), 27-55.
4. M. E. B. BEKKA, On uniqueness of invariant means, Proc. AMS, 126 (1998), 507-514.
5. M. E. B. BEKKA, Restrictions of unitary representations to lattices and associated C*-algebras, J. Funct. Analysis, Vol 143 (1997), 33-41.
6. M. E. B. BEKKA, P-A. CHERIX and P. JOLISSAINT, Kazhdan constants associated with Laplacian on connected Lie groups, J. Lie Theory, 8, no. 1 (1998), 95-110.
7. M. BURGER and P. SARNAK, Ramanujan duals II, Invent. Math., 106, (1991), 1-11.
8. 3(ℤ), J. reine angew. Math., 413 (1991), 36-67.
9. M. E. B. BEKKA and M. MAYER, On Kazhdan's property (T) and Kazhdan constants associated to a Laplacian for SL(3, ℝ), preprint.
10. n(F) where F is a non-archimedian local field, Russian Math. Surveys, 31 (1976), 1-68.
11. 2 matrix coefficients, J. reine angew. Math., 387 (1988), 97-110.
12. 2 groups, Ann. Inst. Fourier, 44-1 (1994), 213-248.
13. M. COWLING, Sur les coefficients des representations unitaires des groupes de Lie simples, Lect. Notes in Math, 739 (1979), 132-178.
14. M. COWLING and T. STEGER, The irreducibility of restrictions of unitary representations to lattices, J. reine angew. Math., 420 (1991), 85-98.
15. J. DIXMIER, C*-Algebras, North-Holland, Amsterdam, 1977.
16. Y. DERRIENNIC and Y. GUIVARC'H, Théorème de renouvellement pour les groupes non moyennables, C. R. Acad. Sci. Paris, 277 (1973), A613-A615.
17. A. DEUTSCH and A. VALETTE, On diameters of orbits of compact groups in unitary representations, J. Austral. Math. Soc., Ser. A, 59 (1995), 308-312.
18. P. EYMARD, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. de France, 92 (1964), 181-236.
19. J. FELL, Weak containment and induced representations of groups, Canad. J. Math., 14 (1962), 237-268.
20. H. FURSTENBERG, Random walks and discrete subgroups of Lie groups, in: Advances in Probability Vol. 1, ed. P. Ney, Marcel Dekker INC, New-York, 1971, 2-63.
21. A. FURMAN and Y. SHALOM, Sharp ergodic theorems for groups actions and strong ergodicity, Ergodic Theory and Dynamical Systems, 19, no. 4 (1999), 1037-1061.
22. A. FURMAN and Y. SHALOM, Random walks on Hilbert spaces and Lyapunov exponents, in preparation.
23. F.P. GREENLEAF, Invariant Means on Topological Groups, Van Nostrand, New-York, 1969.
24. M. GROMOV, Hyperbolic groups, in: Essays in Group Theory, S. Gersten ed., Springer, 1987, 75-265.
25. R. GRIGORCHUK and P. de la HARPE, On problems related to growth, entropy, and spectrum in group theory, J. Dynam. Control Systems, 3 (1997), 51-89.
26. R. GANGOLLI and V.S. VARADARAJAN, Harmonic Analysis of Spherical Functions in Real Reductive Groups, Springer Verlag, 1988.
27. R. E. HOWE and C. C. MOORE, Asymptotic properties of unitary representations, J. Func. Anal, 32 (1979), 72-96.
28. P. de la HARPE, A.G. ROBERTSON, A. VALETTE, On the spectrum of the sum of generators for a finitely generated group, Israel J. of Math., 81, no. 1-2 (1993), 65-96.
29. P. de la HARPE, A.G. ROBERTSON, A. VALETTE, On the spectrum of the sum of generators for a finitely generated group II, Colloq. Math., 65 (1993 vol 1), 87-102.
30. R. HOWE and E.C. TAN, Non-Abelian Harmonic Analysis, Springer Verlag, 1992.
31. P. de la HARPE and A. VALETTE, La Proprieté (T) de Kazhdan pour les Groupes Localement Compacts, Astérisque 175, Société Math. de France, 1989.
32. R. HOWE, On a notion of rank for unitary representations of the classical groups, in: Harmonic Analysis and Group Representations, C.I.M.E., (1982), 223-331.
33. D.A. KAZHDAN, On a connection between the dual space of a group and the structure of its closed subgroups, Func. Anal. Appl., 1 (1967), 63-65.
34. H. KESTEN, Symmetric random walks on groups, Trans. AMS, 92 (1959), 336-354.
35. A.A. KIRILLOV, Elements of the Theory of Representations, Springer Verlag, New York, 1976.
36. D.Y. KLEINBOCK and G.A. MARGULIS, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Amer. Math., Soc. Transl., Ser. 2, 171 (1996), 141-172.
37. D.Y. KLEINBOCK and G.A. MARGULIS, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138, no. 3 (1999), 451-494.
38. A. KNAPP, Representation Theory of Semisimple Groups, Princeton Univ. Press, 1986.
39. B. KOSTANT, On the existence and irreducibility of certain series of representations, Bull. AMS, 75 (1969), 627-642
40. J-S. LI, The minimal decay of matrix coefficients for classical groups, in : Harmonic analysis and its applications in China (1995).
41. A. LUBOTZKY, Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser, 1994.
42. A. LUBOTZKY, Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem, Ann. of Math., 145 (1997), 441-452.
43. 2, II, Comm. Pure and Applied Math., 40 (1987), 401-420.
44. A. LUBOTZKY and B. WEISS, Groups and expanders, in: "Expanding graphs" 95-109, DIMACS series Vol. 10, American Math., Soc., 1993, (Ed: J. Friedman).
45. J. LI, C.B. ZHU, On the Decay of matrix coefficients for exceptional groups, preprint (1995).
46. G.W. MACKEY, Induced representations of locally compact groups, Ann. of Math., 55 (1952), 101-139.
47. G.A. MARGULIS, Discrete Subgroups of Semisimple Groups, Springer Verlag, 1991.
48. C.C. MOORE, Ergodicity of flows on homogeneous spaces, Amer. J. Math., 88 (1966), 154-178.
49. C.C. MOORE, Exponential decay for correlation coefficients for geodesic flows, in : Group representations, ergodic theory, operator algebras and mathematical physics, Conference in honor of G. W. Mackey, MSRI publications (1987), 163-180.
50. A. NEVO, Spectral transfer and pointwise ergodic theorems for semi-simple groups, preprint.
51. H. OH, Tempered subgroups and representations with minimal decay of matrix coefficients, preprint
52. Y. SHALOM, Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan's property (T), Trans, of AMS, (1999), 3387-3412
53. Y. SHALOM, Bounded generation and Kazhdan's property (T), IHES Publ. Math., to appear
54. Y. SHALOM, Random ergodic theorems, invariant means and unitary representations, Tata Inst. Fund. Res. Stud. Math., 14 (Proceedings of the international conference on Lie groups, Bombay 1996) (1998) 273-314.
55. Y. SHALOM, Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math., to appear.
56. p homology, and volumes of hyperbolic 3-manifolds, Trans. AMS, 331, no. 2 (1992), 895-917
57. R.J. ZIMMER, Ergodic Theory and Semisimple Groups, Birkhäuser, 1985.
58. A. ZUK, La propriete (T) de Kazhdan pour les groupes agissant sur les polyèdres, C. R. Acad. Sci. Paris, Ser I, 323, no. 5 (1996), 453-458.
59. A. ZUK, Property (T) and Kazhdan constants for discrete groups, preprint (1999).