Bounded generation and Kazhdan’s property ( T )

Yehuda Shalom

Publications Mathématiques de l'IHÉS (1999)

  • Volume: 90, page 145-168
  • ISSN: 0073-8301

How to cite


Shalom, Yehuda. "Bounded generation and Kazhdan’s property $(T)$." Publications Mathématiques de l'IHÉS 90 (1999): 145-168. <>.

author = {Shalom, Yehuda},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {unitary -representation; topological group; Kazhdan constants; bounded generation; Kazhdan's property},
language = {eng},
pages = {145-168},
publisher = {Institut des Hautes Études Scientifiques},
title = {Bounded generation and Kazhdan’s property $(T)$},
url = {},
volume = {90},
year = {1999},

AU - Shalom, Yehuda
TI - Bounded generation and Kazhdan’s property $(T)$
JO - Publications Mathématiques de l'IHÉS
PY - 1999
PB - Institut des Hautes Études Scientifiques
VL - 90
SP - 145
EP - 168
LA - eng
KW - unitary -representation; topological group; Kazhdan constants; bounded generation; Kazhdan's property
UR -
ER -


  1. [AM] S. I. ADIAN and J. MENNICKE, On bounded generation of SLn(Z), Inter. Jour. Alg. and Comp., Vol. 2, No. 4 (1992), 357-365. Zbl0794.20061MR93m:20064
  2. [BMS] H. BASS, J. MILNOR and J. P. SERRE, Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2), IHES Publ., 33 (1967), 59-137. Zbl0174.05203MR39 #5574
  3. [Bur] M. BURGER, Kazhdan constants for SL3(Z), J. Reine Angew. Math., 413 (1991), 36-67. Zbl0704.22009MR92c:22013
  4. [CK] D. CARTER and G. KELLER, Bounded elementary generation of SLn(O), Amer. J. Math., 105 (1983), 673-687. Zbl0525.20029MR85f:11083
  5. [CW] G. COOKE and P. WEINBERGER, On the constructions of division chains in algebraic number fields, with applications to SL2, Comm Alg., 3 (1975), 481-524. Zbl0315.12001MR52 #8094
  6. [Di] J. DIEUDONNÉ, Sur les fonctions continues p-adiques, Bull. Sci. Math. (2), 68 (1944), 79-95. Zbl0060.08204MR7,111c
  7. [HRV] 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 (1993), No. 1-2, 65-96. Zbl0791.43008MR94j:22007
  8. [HV] P. de La HARPE and A. VALETTE, La Propriété (T) de Kazhdan pour les Groupes Localement Compacts, Astérisque, 175, Société Math. de France (1989). Zbl0759.22001
  9. [Is] R. S. ISMAGILOV, Representations of Infinite-Dimensional Groups, Translations of Mathematical Monographs, 152, Amer. Math. Soc., Providence (1996). Zbl0856.22001MR97g:22014
  10. [Kal] W. van der KALLEN, SL3(C[x]) does not have bounded word length. Proceedings of the conference on Algebraic K-theory (Oberwolfach, 1980), Lecture Notes in Math., 966 (1982), 357-361. Zbl0935.20501MR84b:20037
  11. [Kaz] 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. Zbl0168.27602
  12. [Lub1] A. LUBOTZKY, Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser, 1994. Zbl0826.22012MR96g:22018
  13. [Lub2] A. LUBOTZKY, Subgroup growth and congruence subgroups, Invent. Math., 119 (1995), 267-295. Zbl0848.20036MR95m:20054
  14. [Mok] N. MOK, Harmonic forms with values in locally constant Hilbert bundles, Proceeding of the conference in honor of J. P. Kahane (Orsay 1993), J. Fourier Anal. Appl. (special issue), (1995), 433-453. Zbl0891.58001MR97c:58008
  15. [Mu] V. K. MURTY, Bounded and finite generation of arithmetic groups, Canadian Mathematical Society Conference Proceedings, Vol. 15 (1995), 249-261. Zbl0851.11026MR96g:11039
  16. [MSY] N. MOK, Y. T. SIU and S.-K. YEUNG, Geometric superrigidity, Invent. Math., 113 (1993), 57-83. Zbl0808.53043MR94h:53079
  17. [Pa] P. PANSU, 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), 107-139. Zbl0933.22009MR2000d:53067
  18. [PR] V. PLATONOV and A. RAPINCHUK, Abstract properties of S-arithmetic subgroups and the congruence subgroup problem, Izve. R. Acad. Sci. Ser. Math. (former Math. USSR Izvestiya), 56 (1992), 483-508. Zbl0785.20025
  19. [PS] A. PRESSLEY and G. SEGAL, Loop Groups, Oxford Mathematical Monographs, 1988. Zbl0638.22009
  20. [Rag] M. S. RAGHUNATHAN, On the congruence subgroup problem, IHES Publ., 46 (1976), 107-161. Zbl0347.20027MR58 #22325
  21. [Rap1] A. RAPINCHUK, Congruence subgroup problem for algebraic groups: old and new, Astérisque, 209 (1992), 73-84. Zbl0805.20040MR94b:20047
  22. [Rap2] A. RAPINCHUK, On the finite dimensional unitary representations of Kazhdan groups, Proc. of AMS, 127, No. 5 (1999), 1557-1562. Zbl0926.22001MR99h:22004
  23. [Rap3] A. RAPINCHUK, On SS-rigid groups and A. Weil's criterion for local rigidity I, Manuscripta Math., 97 (1998), 529-543. Zbl0920.20004MR99m:20019
  24. [Sch] W. H. SCHIKHOF, Ultra Calculus: An Introduction to p-Adic Analysis, Cambridge Studies in Advanced Mathematics 4, 1984. Zbl0553.26006MR86j:11104
  25. [Sh1] Y. SHALOM, Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier (Grenoble), 50 (2000) No 3, 833-863. Zbl0966.22004MR2001i:22019
  26. [Sh2] Y. SHALOM, Harmonic maps, superrigidity, and property (T), preprint. 
  27. [Su] A. A. SUSLIN, On the structure of the special linear group over polynomial rings, Math. USSR Izv., Vol. 11 (1977), 221-238. Zbl0378.13002
  28. [Tav] O. I. TAVGEN, Bounded generation of Chevalley groups over rings of algebraic S-integers, Math. USSR Izv., 36 (1991), 101-128. Zbl0709.20024MR91e:20036
  29. [VGG] A. M. VERSHIK, M. I. GRAEV and I. M. GELFAND, Representations of the group SL(2,R), where R is a ring of continuous functions, Russ. Math. Surv., 28 (5) (1973), 87-132. Zbl0297.22003

NotesEmbed ?


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.