## Property $$(T)$$ and rigidity for actions on Banach spaces.(English)Zbl 1162.22005

Classically, a locally compact second countable group $$G$$ has Kazhdan’s property (T) when any unitary representation of $$G$$, with almost invariant vectors, has automatically nontrivial invariant vectors. This is known to be equivalent to Serre’s fixed point property (FH) for affine isometric actions on Hilbert spaces. The work under review undertakes a systematic study of analogue properties (T$$_B$$) and (F$$_B$$) where Hilbert spaces are being replaced by Banach spaces $$B$$.
The first part of the paper investigates relations between properties (T), (T$$_B$$) and (F$$_B$$) for general locally compact second countable groups when $$B=L^p (\mu)$$, with $$\sigma$$-finite measure $$\mu$$ on a standard Borel measure space $$(X,{\mathcal B},\mu)$$. They are summarized in the following two results: 6mm
I.
(T) implies (T$$_B$$) when $$B=L^p(\mu)$$, $$1\leqslant p<\infty$$, or $$B$$ is a closed subspace of $$L^p(\mu)$$, $$1<p<\infty$$, $$p\neq 4,6,8,\ldots$$, or $$B$$ is a quotient space of $$L^p(\mu)$$, $$1<p<\infty$$, $$p\neq \frac{4}{3},\frac{6}{5},\frac{8}{7},\ldots$$ Conversely, if (T$$_{L^p[0,1]}$$) holds for some $$1<p<\infty$$, then (T) holds.
II.
(F$$_B$$) implies (T$$_B$$) for any Banach space $$B$$. In the opposite direction, (T) implies (F$$_B$$) whenever $$B$$ is a closed subspace of $$L^p(\mu)$$, $$1<p< 2+\varepsilon$$, where the constant $$\varepsilon=\varepsilon (G)>0$$ might depend on the group $$G$$.
The second part is concerned with examples of groups that satisfy (F$$_{L^p}$$). It is proved that a large class of higher-rank groups $$G={\mathbf G}_1(k_1)\times \cdots \times {\mathbf G}_m(k_m)$$ and their lattices have property (F$$_B$$) for all Banach spaces $$B$$ mentioned in I. It is also conjectured that these groups should satisfy some analogous properties ($$\overline{\text{T}}_B$$) and ($$\overline{\text{F}}_B$$) for all superreflexive topological vector spaces $$B$$. In this direction, two splitting results about uniformly equicontinuous affine actions on superreflexive spaces of topological group products $$G=G_1\times \cdots \times G_n$$, and respectively of their lattices, are proved. The Appendix contains a generalization, proved by Y. Shalom, of the Howe-Moore theorem on the vanishing of matrix coefficients for unitary representations, to the situation of uniformly convex and uniformly smooth Banach spaces.

### MSC:

 22D40 Ergodic theory on groups 20G15 Linear algebraic groups over arbitrary fields 22D10 Unitary representations of locally compact groups 22D12 Other representations of locally compact groups 22E40 Discrete subgroups of Lie groups 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 53C24 Rigidity results 58D19 Group actions and symmetry properties
Full Text:

### References:

 [1] Alperin, R.: Locally compact groups acting on trees and property T. Monatsh. Math. 93, 261–265 (1982) · Zbl 0488.22014 [2] Banach, S.: Théorie des óperations linéaires. Warsaw, (1932) [3] Bekka, M.E.B.: On uniqueness of invariant means. Proc. Amer. Math. Soc. 126, 507–514 (1998) · Zbl 0885.43003 [4] Bekka, M. E. B., de la Harpe, P., Valette, A.: Kazhdan’s property (T). Preprint (2007) · Zbl 1146.22009 [5] Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society Colloquium Publications, 48. Amer. Math. Soc., Providence, RI (2000) · Zbl 0946.46002 [6] Bergman, G.M.: Generating infinite symmetric groups. Bull. London Math. Soc. 38, 429–440 (2006) · Zbl 1103.20003 [7] 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] · Zbl 0207.03902 [8] 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) [9] 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) [10] Bourdon, M., Pajot, H.: Cohomologie l p et espaces de Besov. J. Reine Angew. Math. 558, 85–108 (2003) · Zbl 1044.20026 [11] Bretagnolle, J., Dacunha-Castelle, D., Krivine, J.L.: Lois stables et espaces L p . Ann. Inst. H. Poincaré Sect. B. 2, 231–259 (1965/1966) · Zbl 0139.33501 [12] Brown, N., Guentner, E.: Uniform embeddings of bounded geometry spaces into reflexive Banach space. Proc. Amer. Math. Soc. 133, 2045–2050 (2005) · Zbl 1069.46003 [13] Burger, M., Monod, N.: Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12, 219–280 (2002) · Zbl 1006.22010 [14] Burger, M., Mozes, S.: Groups acting on trees: from local to global structure. Inst. Hautes Études Sci. Publ. Math. 2000, 113–150 (2001) · Zbl 1007.22012 [15] Cherix, P.A., Cowling, M., Straub, B.: Filter products of Filter products of C o -semigroups and ultraproduct representations for Lie groups. J. Funct. Anal. 208, 31–63 (2004) · Zbl 1060.47043 [16] Connes, A.: A factor of type A factor of type II1 with countable fundamental group. J. Operator Theory 4, 151–153 (1980) · Zbl 0455.46056 [17] Connes, A., Weiss, B.: Property T and asymptotically invariant sequences. Israel J. Math. 37, 209–210 (1980) · Zbl 0479.28017 [18] de Cornulier, Y.: Strongly bounded groups and infinite powers of finite groups. Comm. Algebra 34, 2337–2345 (2006) · Zbl 1125.20023 [19] de Cornulier, Y., Tessera, R. & Valette, A.: Isometric group actions on Banach spaces and representations vanishing at infinity. Preprint, 2006. arXiv:math.RT/0612398 · Zbl 1149.22006 [20] 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) · Zbl 0404.22006 [21] Diestel, J., Uhl Jr., J.J.: Vector Measures Mathematical Surveys, 15. Amer. Math. Soc, Providence, RI (1977) [22] 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) · Zbl 1062.22044 [23] Fisher, D., Margulis, G.A.: Almost isometric actions, property (T), and local rigidity. Invent. Math. 162, 19–80 (2005) · Zbl 1076.22008 [24] 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) · Zbl 1011.46001 [25] Glasner, E., Weiss, B.: Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7, 917–935 (1997) · Zbl 0899.22006 [26] Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, pp. 75–263. Springer, New York (1987) [27] Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13, 73–146 (2003) · Zbl 1122.20021 [28] Guichardet, A.: Sur la cohomologie des groupes topologiques. II. Bull. Sci. Math. 96, 305–332 (1972) · Zbl 0243.57024 [29] Haagerup, U., Przybyszewska, A.: Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces. Preprint, 2006. arXiv:math.OA/0606794 [30] Hardin Jr., C.D.: Isometries on subspaces of L p . Indiana Univ. Math. J. 30, 449–465 (1981) · Zbl 0432.46026 [31] de la Harpe, P., Valette, A.: La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 175 (1989) · Zbl 0759.22001 [32] Higson, N., Lafforgue, V., Skandalis, G.: Counterexamples to the Baum–Connes conjecture. Geom. Funct. Anal. 12, 330–354 (2002) · Zbl 1014.46043 [33] Hjorth, G.: A converse to Dye’s theorem. Trans. Amer. Math. Soc. 357, 3083–3103 (2005) · Zbl 1068.03035 [34] Kakutani, S., Kodaira, K.: Über das Haarsche Mass in der lokal bikompakten Gruppe. Proc. Imp. Acad. Tokyo 20, 444–450 (1944) · Zbl 0060.13501 [35] 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) · Zbl 0168.27602 [36] Klee, V.: Circumspheres and inner products. Math. Scand. 8, 363–370 (1960) · Zbl 0100.31602 [37] Lafforgue, V., Un renforcement de la propriété (T). Preprint (2006) [38] Lamperti, J.: On the isometries of certain function-spaces. Pacific J. Math. 8, 459–466 (1958) · Zbl 0085.09702 [39] Lindenstrauss, J., Tzafriri, L.: On the complemented subspaces problem. Israel J. Math. 9, 263–269 (1971) · Zbl 0211.16301 [40] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, II. Springer, Berlin–Heidelberg (1977, 1979) · Zbl 0362.46013 [41] Lubotzky, A.: Discrete Groups, Expanding Graphs and Invariant Measures. Progress in Mathematics, 125. Birkhäuser, Basel (1994) · Zbl 0826.22012 [42] 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] · Zbl 0312.22011 [43] 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] [44] Margulis, G.A.: Some remarks on invariant means. Monatsh. Math. 90, 233–235 (1980) · Zbl 0436.43002 [45] Margulis, G.A.: On the decomposition of discrete subgroups into amalgams. Selecta Math. Soviet. 1, 197–213 (1981) · Zbl 0515.20031 [46] Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 17. Springer, Berlin–Heidelberg (1991) · Zbl 0732.22008 [47] Mineyev, I.: Straightening and bounded cohomology of hyperbolic groups. Geom. Funct. Anal. 11, 807–839 (2001) · Zbl 1013.20034 [48] Monod, N.: Continuous Bounded Cohomology of Locally Compact Groups. Lecture Notes in Mathematics, 1758. Springer, Berlin–Heidelberg (2001) · Zbl 0967.22006 [49] Monod, N.: Superrigidity for irreducible lattices and geometric splitting. J. Amer. Math. Soc. 19, 781–814 (2006) · Zbl 1105.22006 [50] Monod, N., Shalom, Y.: Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differential Geom. 67, 395–455 (2004) · Zbl 1127.53035 [51] Navas, A.: Actions de groupes de Kazhdan sur le cercle. Ann. Sci. École Norm. Sup. 35, 749–758 (2002) · Zbl 1028.58010 [52] Navas, A.: Reduction of cocycles and groups of diffeomorphisms of the circle. Bull. Belg. Math. Soc. Simon Stevin 13, 193–205 (2006) · Zbl 1131.37043 [53] Pansu, P.: Cohomologie L p : invariance sous quasiisométrie. Preprint (1995) [54] Popa, S.: On the fundamental group of type II1 factors. Proc. Natl. Acad. Sci. USA 101, 723–726 (2004) · Zbl 1064.46048 [55] 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) · Zbl 1120.46043 [56] Popa, S., Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Preprint (2005) arXiv:math.GR/0512646 · Zbl 1131.46040 [57] Pressley, A., Segal, G.: Loop Groups. Oxford Mathematical Monographs. Oxford University Press, New York (1986) · Zbl 0618.22011 [58] Rémy, B.: Integrability of induction cocycles for Kac–Moody groups. Math. Ann. 333, 29–43 (2005) · Zbl 1076.22018 [59] Reznikov, A.: Analytic topology of groups, actions, strings and varietes. Preprint (2000) arXiv:math.DG/0001135 [60] 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) · Zbl 0966.22005 [61] Shalom, Y.: Rigidity of commensurators and irreducible lattices. Invent. Math. 141, 1–54 (2000) · Zbl 0978.22010 [62] 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) · Zbl 0459.28009 [63] Watatani, Y., Property, T.: of Kazhdan implies property FA of Serre. Math. Japon. 27, 97–103 (1982) · Zbl 0489.20022 [64] Wells, J.H., Williams, L.R.: Embeddings and Extensions in Analysis. Springer, New York (1975) · Zbl 0324.46034 [65] Yu, G.: Hyperbolic groups admit proper affine isometric actions on l p -spaces. Geom. Funct. Anal. 15, 1144–1151 (2005) · Zbl 1112.46054 [66] Zimmer, R.J.: Ergodic Theory and Semisimple Groups. Monographs in Mathematics, 81. Birkhäuser, Basel (1984) · Zbl 0571.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.