Abstract
Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in [FM2].
One definition of property (T) is that a group Γ has property (T) if every isometric Γ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with property (T). Some of these are used in the proofs of our local rigidity theorems.
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Abels, H.: Finite presentability of S-arithmetic groups. Compact presentability of solvable groups. Lect. Notes Math., vol. 1261. Berlin: Springer 1987
Bader, U., Furman, A., Gelander, T., Monod, N.: Property (T) and actions on Lp spaces. Preprint
Benjamini, Y., Lindenstrauss, J.: Geometric Non-linear Functional Analysis: Vol. 1. Colloquium Publications. AMS 2000
Benveniste, E.J.: Rigidity of isometric lattice actions on compact Riemannian manifolds. Geom. Funct. Anal. 10, 516–542 (2000)
Bourbaki, N.: Éléments de mathématique. Topologie générale, Chapitres 1 à 4. Paris: Hermann 1971
Candel, A., Conlon, L.: Foliations I. Grad. Stud. Math., vol. 23. Providence, RI: Am. Math. Soc. 2000
do Carmo, M.P.: Riemannian Geometry. Boston: Birkhäuser 1992
Delorme, P.: 1-cohomologie des représentations unitaires des groupes de Lie semi-simples et résolubles. Produits tensoriels continus de représentations (french). Bull. Soc. Math. Fr. 105, 281–336 (1977)
Dieudonné, J.A.E.: Elements d’Analyse, vol. 4. Paris: Gauthiers-Villars 1971
Dolgopyat, D., Krikorian, R.: On simultaneous linearization of diffeomorphisms of the sphere. Preprint
Fisher, D.: First cohomology, rigidity and deformations of isometric group actions. Preprint
Fisher, D., Margulis, G.A.: Local rigidity for cocycles. In: Surv. Differ. Geom., vol. VIII, refereed volume in honor of Calabi, Lawson, Siu and Uhlenbeck, ed. by S.T. Yau, 45 pages. 2003
Fisher, D., Margulis, G.A.: Local rigidity of affine actions of higher rank groups and lattices. Preprint
Ghys, E.: Groupes aléatoire [d’aprés Misha Gromov]. To appear in Semin. Bourbaki
Gromov, M.: Asymptotic invariants of infinite groups. Geometric group theory, vol. 2 (Sussex 1991), pp. 1–295. Lond. Math. Soc. Lect. Note Ser., vol. 182. Cambridge: Cambridge Univ. Press 1993
Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13, 73–146 (2003)
Guichardet, A.: Sur la cohomologie des groupes topologiques. II (french). Bull. Sci. Math. 96, 305–332 (1972)
Harpe, P. de la, Valette, A.: La propriete (T) de Kazhdan pour les groupes localement compacts. Asterisque, vol. 175. Paris: Soc. Math. de France 1989
Heinrich, S.: Ultraproducts in Banach space theory. J. Reine Angew. Math. 313, 72–104 (1980)
Hirsch, M.W.: Differential topology. Grad. Texts Math., No. 33. New York, Heidelberg: Springer 1976
Hörmander, L.: The boundary problems of physical geodesy. Arch. Ration. Mech. Anal. 62, 1–52 (1976)
Kazhdan, D.: On the connection of the dual space of a group with the structure of its closed subgroups (russian). Funct. Anal. Appl. 1, 71–74 (1967)
Karlsson, A., Margulis, G.A.: A multiplicative ergodic theorem and nonpositively curved spaces. Commun. Math. Phys. 208, 107–123 (1999)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. New York: Wiley-Interscience Publications 1963
Margulis, G.A.: Discrete subgroups of semisimple Lie groups. New York: Springer 1991
Markov, A.A.: Three papers on topological groups: I. On the existence of periodic connected topological groups. II. On free topological groups. III. On unconditionally closed sets. Am. Math. Soc. Translation, no. 30., 1950
Mazur, S., Ulam, S.: Sur les transformations isomtriques d’espaces vectoriels normes. C. R. Acad. Sci., Paris 194, 946–948 (1932)
Moore, C.C., Schochet, C.: Global analysis on foliated spaces. With appendices by S. Hurder, Moore, Schochet and R.J. Zimmer. Mathematical Sciences Research Institute Publications, vol. 9. New York: Springer 1988
Mostow, G.D.: Equivariant embeddings in Euclidean space. Ann. Math. (2) 65, 432–446 (1957)
Palais, R.S.: Differential operators on vector bundles. Seminar on the Atiyah-Singer index theorem. Ann. Math. Stud., vol. 57. Princeton: Princeton University Press 1965
Palais, R.S.: Imbedding of compact, differentiable transformation groups in orthogonal representations. J. Math. Mech. 6, 673–678 (1957)
Shalom, Y.: Rigidity of commensurators and irreducible lattices. Invent. Math. 141, 1–54 (2000)
Silberman, L.: Addendum by L. Silberman to “Random walk in random group” by M. Gromov. Geom. Funct. Anal. 13, 147–177 (2003)
van den Dries, L., Wilkie, A.J.: Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra 89, 349–374 (1984)
Zimmer, R.J.: Volume preserving actions of lattices in semisimple groups on compact manifolds. Publ. Math., Inst. Hautes Étud. Sci. 59, 5–33 (1984)
Zimmer, R.J.: Lattices in semisimple groups and distal geometric structures. Invent. Math. 80, 123–137 (1985)
Zimmer, R.J.: Lattices in semisimple groups and invariant geometric structures on compact manifolds. Discrete Groups in Geometry and Analysis: Papers in honor of G.D. Mostow, ed. by R. Howe. Boston: Birkhäuser 1987
Zuk, A.: Property (T) and Kazhdan constants for discrete groups. Geom. Funct. Anal. 13, 643–670 (2003)
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To Yakov G. Sinai on his 70th birthday
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Fisher, D., Margulis, G. Almost isometric actions, property (T), and local rigidity. Invent. math. 162, 19–80 (2005). https://doi.org/10.1007/s00222-004-0437-5
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DOI: https://doi.org/10.1007/s00222-004-0437-5