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Volume preserving actions of lattices in semisimple groups on compact manifolds. (English) Zbl 0576.22013
This paper is the first one in a series studying volume preserving actions of lattices \(\Gamma\) in higher rank semisimple Lie groups on compact manifolds. All presently known actions are of an ”algebraic” nature. The question is whether these are essentially all the examples. One way of exhibiting new actions would be to perturb a given action. One of the main results shows that a sufficiently smooth perturbation of an isometric action is at least topologically isometric.
The proof consists in first showing the existence of a measurable \(\Gamma\)-invariant Riemannian metric - making use of the author’s superrigidity theorem for cocycles which generalizes Margulis’ superrigidity theorem - and then to construct a continuous \(\Gamma\)- invariant Riemannian metric by a Sobolev-lemma type argument.
Reviewer: H.Abels

22E40 Discrete subgroups of Lie groups
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
57S20 Noncompact Lie groups of transformations
37A99 Ergodic theory
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[1] C. Delaroche andA. Kirillov, Sur les relations entre l’espace dual d’un groupe et la structure de ses sousgroupes fermés,Séminaire Bourbaki, No.343 (1967/1968).
[2] Y. Derriennic, Sur le théorème ergodique sous-additif,C. R. Acad. Sci. 281, Paris (1975), Série A, 985–988. · Zbl 0327.60028
[3] H. Furstenberg andH. Kesten, Products of random matrices,Ann. Math. Stat. 31 (1960), 457–469. · Zbl 0137.35501 · doi:10.1214/aoms/1177705909
[4] H. Furstenberg, Rigidity and cocycles for ergodic actions of semisimple groups [after G. A. Margulis and R. Zimmer],Séminaire Bourbaki, No.559 (1979/1980).
[5] S. Helgason,Differential Geometry and Symmetric Spaces, New York, Academic Press, 1962. · Zbl 0111.18101
[6] D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups,Funct. Anal. Appl. 1 (1967), 63–65. · Zbl 0168.27602 · doi:10.1007/BF01075866
[7] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes,J. Royal Stat. Soc. B30 (1968), 499–510. · Zbl 0182.22802
[8] S. Kobayashi,Transformation groups in differential geometry, New York, Springer, 1972. · Zbl 0246.53031
[9] G. A. Margulis, Discrete groups of motions of manifolds of non-positive curvature,A.M.S. Translations 109 (1977), 33–45. · Zbl 0367.57012
[10] D. Montgomery andL. Zippin,Topological Transformation Groups, New York, Interscience, 1955.
[11] G. D. Mostow, Strong rigidity of locally symmetric spaces,Annals of Math. Studies, No.78 (1973). · Zbl 0265.53039
[12] G. D. Mostow, Intersection of discrete subgroups with Cartan subgroups,J. Indian Math. Soc. 34 (1970), 203–214. · Zbl 0235.22019
[13] R. Palais, Seminar on the Atiyah-Singer Index Theorem,Annals of Math. Studies, No.57. · Zbl 0202.23103
[14] Ya. B. Pesin, Lyapunov characteristic exponents and smooth ergodic theory,Russian Math. Surveys 32 (1977), 55–114. · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639
[15] G. Prasad andM. S. Raghunathan, Cartan subgroups and lattices in semisimple groups,Annals of Math. 96 (1972), 296–317. · Zbl 0245.22013 · doi:10.2307/1970790
[16] M. S. Raghunathan, On the congruence subgroup problem,Publ. Math. I.H.E.S. 46 (1976), 107–161. · Zbl 0347.20027
[17] K. Schmidt, Amenability, Kazhdan’s property T, strong ergodicity, and invariant means for group actions,Erg. Th. and Dyn. Sys. 1 (1981), 223–236. · Zbl 0485.28019
[18] S. P. Wang, On isolated points in the dual spaces of locally compact groups,Math. Ann. 218 (1975), 19–34. · Zbl 0332.22009 · doi:10.1007/BF01350065
[19] A. Weil, On discrete subgroups of Lie groups, I, II,Annals of Math. 72 (1960), 369–384, andibid.,75 (1962), 578–602. · Zbl 0131.26602 · doi:10.2307/1970140
[20] D. Zelobenko, Compact Lie groups and their representations,Translations of Math. Monographs, vol. 40, A.M.S., Providence, R.I., 1973. · Zbl 0272.22006
[21] R. J. Zimmer, Extensions of ergodic group actions,Ill. J. Math. 20 (1976), 373–409. · Zbl 0334.28015
[22] R. J. Zimmer, Amenable pairs of ergodic actions and the associated von Neumann algebras,Trans. A.M.S. 243 (1978), 271–286. · Zbl 0408.22011 · doi:10.1090/S0002-9947-1978-0502907-6
[23] R. J. Zimmer, Orbit spaces of unitary representations, ergodic theory, and simple Lie groups,Annals of Math. 106 (1977), 573–588. · Zbl 0393.22006 · doi:10.2307/1971068
[24] R. J. Zimmer, An algebraic group associated to an ergodic diffeomorphism,Comp. Math. 43 (1981), 59–69. · Zbl 0491.58020
[25] R. J. Zimmer, On the cohomology of ergodic actions of semisimple Lie groups and discrete subgroups,Amer. J. Math. 103 (1981), 937–950. · Zbl 0475.22011 · doi:10.2307/2374253
[26] R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups,Annals of Math. 112 (1980), 511–529. · Zbl 0468.22011 · doi:10.2307/1971090
[27] R. J. Zimmer, Orbit equivalence and rigidity of ergodic actions of Lie groups,Erg. Th. and Dyn. Sys. 1 (1981), 237–253. · Zbl 0485.22013
[28] R. J. Zimmer, Ergodic theory, group representations, and rigidity,Bull. A.M.S. 6 (1982), 383–416. · Zbl 0532.22009 · doi:10.1090/S0273-0979-1982-15005-4
[29] R. J. Zimmer,Ergodic Theory and Semisimple Groups, forthcoming. · Zbl 0571.58015
[30] R. J. Zimmer, Arithmetic groups acting on compact manifolds,Bull. A.M.S., to appear. · Zbl 0532.22012
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