Nouvelles approches de la propriété (T) de Kazhdan

Valette, Alain

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 913, pp. 97-124.

Un groupe localement compact $G$ a la propriété (T) de Kazhdan si la $1$-cohomologie de tout $G$-module hilbertien est nulle. Cette propriété de rigidité de la théorie des représentations de $G$ a trouvé des applications qui vont de la théorie ergodique à la théorie des graphes. Pendant près de 30 ans, les seuls exemples connus de groupes avec la propriété (T), provenaient des groupes algébriques simples sur les corps locaux, ou de leurs réseaux. La situation a radicalement changé ces dernières années : nouvelles caractérisations (Y. Shalom), nouveaux exemples (M. Gromov, Y. Shalom, A. Zuk), de sorte qu’on peut même parler de “généricité” des groupes discrets ayant la propriété (T).

A locally compact group $G$ has Kazhdan’s property (T) if the 1-cohomology of any unitary $G$-module is zero. This rigidity property of the representation theory of $G$ found applications ranging from ergodic theory to graph theory. For nearly 30 years, the only known examples of groups with property (T) came from simple algebraic groups over local fields, and their lattices. Situation dramatically changed during the last years: new characterizations (Y. Shalom), new examples (M. Gromov, Y. Shalom, A. Zuk), so that one may talk of “genericity” of discrete groups with property (T).

[1] R. Alperin - “Locally compact groups acting on trees and property (T)”, Monatsh. Math. 93 (1982), p. 261-265. | MR | Zbl

[2] W. Ballmann & J. Swiatkowski - “On $L^2$-cohomology and property (T) for automorphism groups of polyhedral cell complexes”, Geom. funct. anal. 7 (1997), p. 615-645. | MR | Zbl

[3] S. Barré - “Immeubles de Tits triangulaires exotiques”, Ann. Fac. Sci. Toulouse Math. (5) 9 (2000), p. 575-603. | Numdam | MR | Zbl

[4] H. Behr - “$S{L}_3\left(F_q\left[t\right]\right)$ is not finitely presentable”, in Homological and Combinational Techniques in Group Theory, Proc. Symp., Durham, 1977, London Math. Soc. Lect. Notes Ser., vol. 36, 1979, p. 213-224. | MR | Zbl

[5] M. E. B. Bekka, P. De La Harpe & A. Valette - “Kazhdan's property (T)”, Preprint, septembre 2003, http://www.unige.ch/math/biblio/preprint/2003/tsept03.ps. | MR

[6] M. E. B. Bekka & A. Valette - “Group cohomology, harmonic functions and the first $L^2$-Betti number”, Potential analysis 6 (1997), p. 313-326. | MR | Zbl

[7] A. Borel - “Cohomologie de certains groupes discrets et laplacien $p$-adique (d’après H. Garland)”, in Sém. Bourbaki (1973-74), Lect. Notes in Math., vol. 431, Springer-Verlag, 1975, exp. no 437, p. 12-35. | Numdam | MR | Zbl

[8] F. Bruhat & J. Tits - “Groupes réductifs sur un corps local I ; données radicielles valuées”, Inst. Hautes Études Sci. Publ. Math. 41 (1972), p. 5-251. | Numdam | MR | Zbl

[9] M. Burger & N. Monod - “Bounded cohomology of lattices in higher rank Lie groups”, J. Eur. Math. Soc. 1 (1999), p. 199-235. | MR | Zbl

[10] D. Cartwright, A. Mantero, T. Steger & A. Zappa - “Groups acting simply transitively on the vertices of a building of type ${\tilde{A}}_2$”, Geom. Dedicata 47 (1993), p. 143-166. | MR | Zbl

[11] D. Cartwright, W. Mlotkowski & T. Steger - “Property (T) and ${\tilde{A}}_2$ groups”, Ann. Inst. Fourier (Grenoble) 44 (1993), p. 213-248. | Numdam | MR | Zbl

[12] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg & A. Valette - Groups with the Haagerup property (Gromov's a-T-menability), Progress in Math., Birkhäuser, 2001. | MR | Zbl

[13] K. Corlette - “Archimedean superrigidity and hyperbolic rigidity”, Ann. of Math. 135 (1992), p. 165-182. | MR | Zbl

[14] M. Cowling - “Sur les coefficients des représentations des groupes de Lie simples”, in Analyse harmonique sur les groupes de Lie II (Sém. Nancy-Strasbourg 1976-1978), Lect. Notes in Math., vol. 739, Springer, 1979, p. 132-178. | MR | Zbl

[15] C. Delaroche & A. Kirillov - “Sur les relations entre l'espace dual d'un groupe et la structure de ses sous-groupes fermés (d'après D.A. Kajdan)”, in Sém. Bourbaki (1967-68), W. A. Benjamin, Inc., New York-Amsterdam, 1969, exp. no 343 ou Soc. Math. de France, Collection Hors Série, vol. 10, (1995), en réimpression, p. 507-528. | Numdam | MR | Zbl

[16] P. Delorme - “$1$-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles”, Bull. Soc. Math. France 105 (1977), p. 281-336. | Numdam | MR | Zbl

[17] J. Faraut - “Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques”, in Analyse harmonique, Cours du CIMPA, Nice, 1983, p. 315-446. | Zbl

[18] D. Farley - “Proper isometric actions of Thompson's groups on Hilbert space”, Preprint, 2002. | MR | Zbl

[19] W. Feit & G. Higman - “The nonexistence of certain generalized polygons”, J. Algebra 1 (1964), p. 114-131. | MR | Zbl

[20] A. Furman - “Gromov's measure equivalence and rigidity of higher rank lattices”, Ann. of Math. 150 (1999), p. 1059-1081. | MR | Zbl

[21] H. Garland - “$p$-adic curvature and the cohomology of discrete subgroups of $p$-adic groups”, Ann. of Math. 97 (1973), p. 375-423. | MR | Zbl

[22] T. Gelander & A. Zuk - “Dependence of Kazhdan constants on generating subsets”, Israel J. Math. 129 (2002), p. 93-98. | MR | Zbl

[23] É. Ghys - “Actions de réseaux sur le cercle”, Invent. Math. 137 (1999), p. 199-231. | MR | Zbl

[24] É. Ghys & V. Sergiescu - “Sur un groupe remarquable de difféomorphismes du cercle”, Comment. Math. Helv. 62 (1987), p. 185-239. | MR | Zbl

[25] M. Gromov - “Hyperbolic groups”, in Essays in group theory (S.M. Gersten, 'ed.), vol. 8, Math. Sci. Res. Inst. Publ., Springer-Verlag, New York, 1987, p. 75-263. | MR | Zbl

[26] -, “Random walk in random groups”, 13 (2003), p. 73-146. | MR

[27] M. Gromov & R. Schoen - “Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one”, Inst. Hautes Études Sci. Publ. Math. 76 (1992), p. 165-246. | Numdam | MR | Zbl

[28] A. Guichardet - “Sur la cohomologie des groupes topologiques II”, Bull. Sci. Math. 96 (1972), p. 305-332. | MR | Zbl

[29] -, Cohomologie des groupes topologiques et des algèbres de Lie, Cedic-F. Nathan, 1980. | Zbl

[30] P. De La Harpe & A. Valette - La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque, vol. 175, Soc. Math. France, Paris, 1989. | Numdam | Zbl

[31] G. Hector & U. Hirsch - The geometry of foliations, Part B, Vieweg, 1983. | MR

[32] S. Helgason - Differential geometry and symmetric spaces, Academic Press, 1963. | MR | Zbl

[33] A. Hulanicki - “Means and Følner condition on locally compact groups”, Studia Math. 24 (1966), p. 87-104. | MR | Zbl

[34] J. Jost - “Equilibrium maps between metric spaces”, Calc. Var. 2 (1994), p. 173-204. | MR | Zbl

[35] D. Kazhdan - “Connection of the dual space of a group with the structure of its closed subgroups”, Funct. Anal. Appl. 1 (1967), p. 63-65. | MR | Zbl

[36] B. Kostant - “On the existence and irreducibility of certain series of representations”, Bull. Amer. Math. Soc. 75 (1969), p. 627-642. | MR | Zbl

[37] A. Lubotzky - Discrete groups, expanding graphs and invariant measures, Progress in Math., vol. 125, Birkhäuser Verlag, Basel, 1994. | MR | Zbl

[38] G. A. Margulis - Discrete subgroups of semisimple Lie groups, Frgeb. Math. Grenzgeb. 3 Folge, vol. 17, Springer-Verlag, 1991. | MR | Zbl

[39] N. Mok - “Harmonic forms with values in locally constant Hilbert bundles”, J. Fourier analysis and appl. (1995), p. 433-453, Proceedings of the Conference in honor of J.-P. Kahane (Orsay 1993), Special Volume. | MR | Zbl

[40] A. Navas - “Actions de groupes de Kazhdan sur le cercle”, Ann. Sci. École Norm. Sup. (4) 35 (2002), p. 749-758. | Numdam | MR | Zbl

[41] P. Pansu - “Sous-groupes discrets des groupes de Lie : rigidité, arithméticité”, in Sém. Bourbaki (1993-94), Astérisque, vol. 227, Soc. Math. France, Paris, 1995, exp. no 778, p. 69-105. | Numdam | MR | Zbl

[42] -, “Formules de Matsushima, de Garland, et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles”, Bull. Soc. Math. France 126 (1998), p. 107-139. | Numdam | MR | Zbl

[43] A. Pressley & G. Segal - Loop groups, Oxford Univ. Press, 1986. | MR | Zbl

[44] U. Rehmann & C. Soulé - “Finitely presented groups of matrices”, in Algebraic K-theory, Lect. Notes in Math., vol. 551, Springer, 1976, p. 164-169. | MR | Zbl

[45] H. Reiter - “Some properties of locally compact groups”, Nederl. Akad. Wetensch. Indag. Math. 27 (1965), p. 697-701. | MR | Zbl

[46] A. Reznikov - “Analytic topology of groups, actions, strings and varieties”, Preprint, janvier 2000. | MR | Zbl

[47] J.-P. Serre - Arbres, amalgames, ${SL}_2$, Astérisque, vol. 46, Soc. Math. France, Paris, 1977. | Numdam | MR | Zbl

[48] Y. Shalom - “Rigidity of commensurators and irreducible lattices”, Invent. Math. 141 (2000), p. 1-54. | MR | Zbl

[49] -, “Bounded generation and Kazhdan property (T)”, Inst. Hautes Études Sci. Publ. Math. 90 (2001), p. 145-168. | Numdam | Zbl

[50] G. Skandalis - “Une notion de nucléarité en $K$-théorie”, $K$-Theory 1 (1988), p. 549-573. | MR | Zbl

[51] L. N. Vaserstein - “Groups having the property (T)”, Funct. Anal. Appl. 2 (1968), p. 174. | MR | Zbl

[52] A. M. Vershik & S. I. Karpushev - “Cohomology of groups in unitary representations, the neighbourhood of the identity and conditionally positive definite functions”, Math. USSR-Sb. 47 (1984), p. 513-526. | Zbl

[53] S. P. Wang - “The dual space of semi-simple Lie groups”, Amer. J. Math. 23 (1969), p. 921-937. | MR | Zbl

[54] Y. Watatani - “Property (T) of Kazhdan implies property (FA) of Serre”, Math. Japon. 27 (1981), p. 97-103. | MR | Zbl

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

[56] -, “Property (T) and Kazhdan constants for discrete groups”, Geom. Funct. Anal. 13 (2003), p. 643-670. | MR | Zbl