New approaches to Kazhdan’s property (T)

Alain Valette

Séminaire Bourbaki (2002-2003)

  • Volume: 45, page 97-124
  • ISSN: 0303-1179

Abstract

top
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).

How to cite

top

Valette, Alain. "Nouvelles approches de la propriété (T) de Kazhdan." Séminaire Bourbaki 45 (2002-2003): 97-124. <http://eudml.org/doc/252136>.

@article{Valette2002-2003,
abstract = {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).},
author = {Valette, Alain},
journal = {Séminaire Bourbaki},
keywords = {unitary representations; 1-cohomology; simple algebraic groups; lattices; harmonic maps; graph spectra},
language = {fre},
pages = {97-124},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Nouvelles approches de la propriété (T) de Kazhdan},
url = {http://eudml.org/doc/252136},
volume = {45},
year = {2002-2003},
}

TY - JOUR
AU - Valette, Alain
TI - Nouvelles approches de la propriété (T) de Kazhdan
JO - Séminaire Bourbaki
PY - 2002-2003
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 45
SP - 97
EP - 124
AB - 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).
LA - fre
KW - unitary representations; 1-cohomology; simple algebraic groups; lattices; harmonic maps; graph spectra
UR - http://eudml.org/doc/252136
ER -

References

top
  1. [1] R. Alperin – “Locally compact groups acting on trees and property (T)”, Monatsh. Math.93 (1982), p. 261–265. Zbl0488.22014MR666827
  2. [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. Zbl0897.22007MR1465598
  3. [3] S. Barré – “Immeubles de Tits triangulaires exotiques”, Ann. Fac. Sci. Toulouse Math. (5) 9 (2000), p. 575–603. Zbl1003.51007MR1838139
  4. [4] H. Behr – “ S L 3 ( F q [ t ] ) 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. Zbl0434.20025MR564424
  5. [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. MR1995797
  6. [6] M.E.B. Bekka & A. Valette – “Group cohomology, harmonic functions and the first L 2 -Betti number”, Potential analysis6 (1997), p. 313–326. Zbl0882.22013MR1452785
  7. [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. Zbl0376.22009MR476919
  8. [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. Zbl0254.14017MR327923
  9. [9] M. Burger & N. Monod – “Bounded cohomology of lattices in higher rank Lie groups”, J. Eur. Math. Soc.1 (1999), p. 199–235. Zbl0932.22008MR1694584
  10. [10] D. Cartwright, A. Mantero, T. Steger & A. Zappa – “Groups acting simply transitively on the vertices of a building of type A ˜ 2 ”, Geom. Dedicata47 (1993), p. 143–166. Zbl0784.51010MR1232965
  11. [11] D. Cartwright, W. Mlotkowski & T. Steger – “Property (T) and A ˜ 2 groups”, Ann. Inst. Fourier (Grenoble) 44 (1993), p. 213–248. Zbl0792.43002MR1262886
  12. [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. Zbl1030.43002MR1852148
  13. [13] K. Corlette – “Archimedean superrigidity and hyperbolic rigidity”, Ann. of Math.135 (1992), p. 165–182. Zbl0768.53025MR1147961
  14. [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. Zbl0417.22010MR560837
  15. [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. Zbl0214.04602MR1610473
  16. [16] P. Delorme – “ 1 -cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles”, Bull. Soc. Math. France105 (1977), p. 281–336. Zbl0404.22006MR578893
  17. [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. Zbl0569.43002
  18. [18] D. Farley – “Proper isometric actions of Thompson’s groups on Hilbert space”, Preprint, 2002. Zbl1113.22005MR2006480
  19. [19] W. Feit & G. Higman – “The nonexistence of certain generalized polygons”, J. Algebra1 (1964), p. 114–131. Zbl0126.05303MR170955
  20. [20] A. Furman – “Gromov’s measure equivalence and rigidity of higher rank lattices”, Ann. of Math.150 (1999), p. 1059–1081. Zbl0943.22013MR1740986
  21. [21] H. Garland – “ p -adic curvature and the cohomology of discrete subgroups of p -adic groups”, Ann. of Math.97 (1973), p. 375–423. Zbl0262.22010MR320180
  22. [22] T. Gelander & A. Zuk – “Dependence of Kazhdan constants on generating subsets”, Israel J. Math.129 (2002), p. 93–98. Zbl0993.22003MR1910934
  23. [23] É. Ghys – “Actions de réseaux sur le cercle”, Invent. Math.137 (1999), p. 199–231. Zbl0995.57006MR1703323
  24. [24] É. Ghys & V. Sergiescu – “Sur un groupe remarquable de difféomorphismes du cercle”, Comment. Math. Helv.62 (1987), p. 185–239. Zbl0647.58009MR896095
  25. [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. Zbl0634.20015MR919829
  26. [26] —, “Random walk in random groups”, 13 (2003), p. 73–146. Zbl1122.20021MR1978492
  27. [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. Zbl0896.58024MR1215595
  28. [28] A. Guichardet – “Sur la cohomologie des groupes topologiques II”, Bull. Sci. Math.96 (1972), p. 305–332. Zbl0243.57024MR340464
  29. [29] —, Cohomologie des groupes topologiques et des algèbres de Lie, Cedic–F. Nathan, 1980. Zbl0464.22001
  30. [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. Zbl0759.22001
  31. [31] G. Hector & U. Hirsch – The geometry of foliations, Part B, Vieweg, 1983. MR726931
  32. [32] S. Helgason – Differential geometry and symmetric spaces, Academic Press, 1963. Zbl0111.18101MR145455
  33. [33] A. Hulanicki – “Means and Følner condition on locally compact groups”, Studia Math.24 (1966), p. 87–104. Zbl0165.48701MR195982
  34. [34] J. Jost – “Equilibrium maps between metric spaces”, Calc. Var.2 (1994), p. 173–204. Zbl0798.58021MR1385525
  35. [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. Zbl0168.27602MR209390
  36. [36] B. Kostant – “On the existence and irreducibility of certain series of representations”, Bull. Amer. Math. Soc.75 (1969), p. 627–642. Zbl0229.22026MR245725
  37. [37] A. Lubotzky – Discrete groups, expanding graphs and invariant measures, Progress in Math., vol. 125, Birkhäuser Verlag, Basel, 1994. Zbl0826.22012MR1308046
  38. [38] G.A. Margulis – Discrete subgroups of semisimple Lie groups, Frgeb. Math. Grenzgeb. 3 Folge, vol. 17, Springer-Verlag, 1991. Zbl0732.22008MR1090825
  39. [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. Zbl0891.58001MR1364901
  40. [40] A. Navas – “Actions de groupes de Kazhdan sur le cercle”, Ann. Sci. École Norm. Sup. (4) 35 (2002), p. 749–758. Zbl1028.58010MR1951442
  41. [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. Zbl0835.22011MR1321644
  42. [42] —, “Formules de Matsushima, de Garland, et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles”, Bull. Soc. Math. France126 (1998), p. 107–139. Zbl0933.22009MR1651383
  43. [43] A. Pressley & G. Segal – Loop groups, Oxford Univ. Press, 1986. Zbl0638.22009MR900587
  44. [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. Zbl0445.20025MR486175
  45. [45] H. Reiter – “Some properties of locally compact groups”, Nederl. Akad. Wetensch. Indag. Math.27 (1965), p. 697–701. Zbl0131.13201MR194908
  46. [46] A. Reznikov – “Analytic topology of groups, actions, strings and varieties”, Preprint, janvier 2000. Zbl1196.57023MR2402403
  47. [47] J.-P. Serre – Arbres, amalgames, S L 2 , Astérisque, vol. 46, Soc. Math. France, Paris, 1977. Zbl0369.20013MR476875
  48. [48] Y. Shalom – “Rigidity of commensurators and irreducible lattices”, Invent. Math.141 (2000), p. 1–54. Zbl0978.22010MR1767270
  49. [49] —, “Bounded generation and Kazhdan property (T)”, Inst. Hautes Études Sci. Publ. Math.90 (2001), p. 145–168. Zbl0980.22017
  50. [50] G. Skandalis – “Une notion de nucléarité en K -théorie”, K -Theory 1 (1988), p. 549–573. Zbl0653.46065MR953916
  51. [51] L.N. Vaserstein – “Groups having the property (T)”, Funct. Anal. Appl. 2 (1968), p. 174. Zbl0232.20087MR228620
  52. [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. Zbl0528.43005
  53. [53] S.P. Wang – “The dual space of semi-simple Lie groups”, Amer. J. Math.23 (1969), p. 921–937. Zbl0192.36102MR259023
  54. [54] Y. Watatani – “Property (T) of Kazhdan implies property (FA) of Serre”, Math. Japon.27 (1981), p. 97–103. Zbl0489.20022MR649023
  55. [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. Zbl0858.22007MR1408975
  56. [56] —, “Property (T) and Kazhdan constants for discrete groups”, Geom. Funct. Anal.13 (2003), p. 643–670. Zbl1036.22004MR1995802

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.