Acta Mathematica

Property (T) and rigidity for actions on Banach spaces

Uri Bader, Alex Furman, Tsachik Gelander, and Nicolas Monod

Full-text: Open access


We study property (T) and the fixed-point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for Lp follows from property (T) when 1< p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for Lp holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces.


Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).

Article information

Acta Math., Volume 198, Number 1 (2007), 57-105.

Received: 2 August 2005
Accepted: 5 February 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2007 © Institut Mittag-Leffler


Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas. Property ( T ) and rigidity for actions on Banach spaces. Acta Math. 198 (2007), no. 1, 57--105. doi:10.1007/s11511-007-0013-0.

Export citation


  • Alperin, R.: Locally compact groups acting on trees and property T. Monatsh. Math. 93, 261–265 (1982)
  • Banach, S.: Théorie des óperations linéaires. Warsaw, (1932)
  • Bekka, M.E.B.: On uniqueness of invariant means. Proc. Amer. Math. Soc. 126, 507–514 (1998)
  • Bekka, M. E. B., de la Harpe, P., Valette, A.: Kazhdan’s property (T). Preprint (2007)
  • Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society Colloquium Publications, 48. Amer. Math. Soc., Providence, RI (2000)
  • Bergman, G.M.: Generating infinite symmetric groups. Bull. London Math. Soc. 38, 429–440 (2006)
  • Bernshtein, I.N., Kazhdan, D.A.: The one-dimensional cohomology of discrete subgroups. Funktsional. Anal. i Prilozhen. 4(1), 1–5 (1970) [(Russian); English translation in Funct. Anal. Appl., 4 (1970), 1–4]
  • Bourbaki, N.: Éléments de mathématique. Fasc. X. Première partie. Livre III: Topologie générale. Chapitre 10: Espaces fonctionnels. Deuxième édition, entièrement refondue. Actualités Sci. Indust., No. 1084. Hermann, Paris (1961)
  • Bourbaki, N.: Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations. Actualités Scientifiques et Industrielles, No. 1306. Hermann, Paris, (1963)
  • Bourdon, M., Pajot, H.: Cohomologie lp et espaces de Besov. J. Reine Angew. Math. 558, 85–108 (2003)
  • Bretagnolle, J., Dacunha-Castelle, D., Krivine, J.L.: Lois stables et espaces Lp. Ann. Inst. H. Poincaré Sect. B. 2, 231–259 (1965/1966)
  • Brown, N., Guentner, E.: Uniform embeddings of bounded geometry spaces into reflexive Banach space. Proc. Amer. Math. Soc. 133, 2045–2050 (2005)
  • Burger, M., Monod, N.: Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12, 219–280 (2002)
  • Burger, M., Mozes, S.: Groups acting on trees: from local to global structure. Inst. Hautes Études Sci. Publ. Math. 2000, 113–150 (2001)
  • Cherix, P.A., Cowling, M., Straub, B.: Filter products of Filter products of Co-semigroups and ultraproduct representations for Lie groups. J. Funct. Anal. 208, 31–63 (2004)
  • Connes, A.: A factor of type A factor of type II1 with countable fundamental group. J. Operator Theory 4, 151–153 (1980)
  • Connes, A., Weiss, B.: Property T and asymptotically invariant sequences. Israel J. Math. 37, 209–210 (1980)
  • de Cornulier, Y.: Strongly bounded groups and infinite powers of finite groups. Comm. Algebra 34, 2337–2345 (2006)
  • de Cornulier, Y., Tessera, R. & Valette, A.: Isometric group actions on Banach spaces and representations vanishing at infinity. Preprint, 2006.
  • Delorme, P.: 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations. Bull. Soc. Math. France 105, 281–336 (1977)
  • Diestel, J., Uhl Jr., J.J.: Vector Measures Mathematical Surveys, 15. Amer. Math. Soc, Providence, RI (1977)
  • Fisher, D., Margulis, G. A.: Local rigidity for cocycles. In: Surveys in Differential Geometry, Vol. VIII (Boston, MA, 2002), pp. 191–234. International Press, Somerville, MA (2003)
  • Fisher, D., Margulis, G.A.: Almost isometric actions, property (T), and local rigidity. Invent. Math. 162, 19–80 (2005)
  • Fleming, R.J., Jamison, J.E.: Isometries on Banach Spaces: Function Spaces. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129. Chapman & Hall/CRC, Boca Raton, FL (2003)
  • Glasner, E., Weiss, B.: Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7, 917–935 (1997)
  • Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, pp. 75–263. Springer, New York (1987)
  • Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13, 73–146 (2003)
  • Guichardet, A.: Sur la cohomologie des groupes topologiques. II. Bull. Sci. Math. 96, 305–332 (1972)
  • Haagerup, U., Przybyszewska, A.: Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces. Preprint, 2006.
  • Hardin Jr., C.D.: Isometries on subspaces of Lp. Indiana Univ. Math. J. 30, 449–465 (1981)
  • de la Harpe, P., Valette, A.: La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 175 (1989)
  • Higson, N., Lafforgue, V., Skandalis, G.: Counterexamples to the Baum–Connes conjecture. Geom. Funct. Anal. 12, 330–354 (2002)
  • Hjorth, G.: A converse to Dye’s theorem. Trans. Amer. Math. Soc. 357, 3083–3103 (2005)
  • Kakutani, S., Kodaira, K.: Über das Haarsche Mass in der lokal bikompakten Gruppe. Proc. Imp. Acad. Tokyo 20, 444–450 (1944)
  • Kazhdan, D.A.: Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. Appl. 1, 63–65 (1967)
  • Klee, V.: Circumspheres and inner products. Math. Scand. 8, 363–370 (1960)
  • Lafforgue, V., Un renforcement de la propriété (T). Preprint (2006)
  • Lamperti, J.: On the isometries of certain function-spaces. Pacific J. Math. 8, 459–466 (1958)
  • Lindenstrauss, J., Tzafriri, L.: On the complemented subspaces problem. Israel J. Math. 9, 263–269 (1971)
  • Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, II. Springer, Berlin–Heidelberg (1977, 1979)
  • Lubotzky, A.: Discrete Groups, Expanding Graphs and Invariant Measures. Progress in Mathematics, 125. Birkhäuser, Basel (1994)
  • Margulis, G.A.: Explicit constructions of expanders. Problemy Peredachi Informacii 9(4), 71–80 (1973) [(Russian); English translation in Problems Inform. Transmission, 9 (1973), 325–332]
  • Margulis, G.A.: Finiteness of quotient groups of discrete subgroups. Funktsional. Anal. i Prilozhen. 13, 28–39 (1979) [(Russian); English translation in Funct. Anal. Appl., 13 (1979), 178–187]
  • Margulis, G.A.: Some remarks on invariant means. Monatsh. Math. 90, 233–235 (1980)
  • Margulis, G.A.: On the decomposition of discrete subgroups into amalgams. Selecta Math. Soviet. 1, 197–213 (1981)
  • Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 17. Springer, Berlin–Heidelberg (1991)
  • Mineyev, I.: Straightening and bounded cohomology of hyperbolic groups. Geom. Funct. Anal. 11, 807–839 (2001)
  • Monod, N.: Continuous Bounded Cohomology of Locally Compact Groups. Lecture Notes in Mathematics, 1758. Springer, Berlin–Heidelberg (2001)
  • Monod, N.: Superrigidity for irreducible lattices and geometric splitting. J. Amer. Math. Soc. 19, 781–814 (2006)
  • Monod, N., Shalom, Y.: Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differential Geom. 67, 395–455 (2004)
  • Navas, A.: Actions de groupes de Kazhdan sur le cercle. Ann. Sci. École Norm. Sup. 35, 749–758 (2002)
  • Navas, A.: Reduction of cocycles and groups of diffeomorphisms of the circle. Bull. Belg. Math. Soc. Simon Stevin 13, 193–205 (2006)
  • Pansu, P.: Cohomologie Lp: invariance sous quasiisométrie. Preprint (1995)
  • Popa, S.: On the fundamental group of type II1 factors. Proc. Natl. Acad. Sci. USA 101, 723–726 (2004)
  • Popa, S.: Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. I, II. Invent. Math. 165, 369–408, 409–451 (2006)
  • Popa, S., Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Preprint (2005)
  • Pressley, A., Segal, G.: Loop Groups. Oxford Mathematical Monographs. Oxford University Press, New York (1986)
  • Rémy, B.: Integrability of induction cocycles for Kac–Moody groups. Math. Ann. 333, 29–43 (2005)
  • Reznikov, A.: Analytic topology of groups, actions, strings and varietes. Preprint (2000)
  • Robertson, G., Steger, T.: Negative definite kernels and a dynamical characterization of property (T) for countable groups. Ergodic Theory Dynam. Systems 18, 247–253 (1998)
  • Shalom, Y.: Rigidity of commensurators and irreducible lattices. Invent. Math. 141, 1–54 (2000)
  • Sullivan, D.: For n>3 there is only one finitely additive rotationally invariant measure on the n-sphere defined on all Lebesgue measurable subsets. Bull. Amer. Math. Soc. 4, 121–123 (1981)
  • Watatani, Y., Property, T.: of Kazhdan implies property FA of Serre. Math. Japon. 27, 97–103 (1982)
  • Wells, J.H., Williams, L.R.: Embeddings and Extensions in Analysis. Springer, New York (1975)
  • Yu, G.: Hyperbolic groups admit proper affine isometric actions on lp-spaces. Geom. Funct. Anal. 15, 1144–1151 (2005)
  • Zimmer, R.J.: Ergodic Theory and Semisimple Groups. Monographs in Mathematics, 81. Birkhäuser, Basel (1984)