Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T09:52:49.711Z Has data issue: false hasContentIssue false
  • English
  • Français

Orbit equivalence and rigidity of ergodic actions of Lie groups

Published online by Cambridge University Press:  19 September 2008

Robert J. Zimmer
Robert J. Zimmer*
Affiliation:
From the Department of Mathematics, University of Chicago, USA
*
†Address for correspondence: Dr Robert J. Zimmer, Department of Mathematics, University of Chicago, Chicago, III. 60637, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 1 , Issue 2 , June 1981 , pp. 237 - 253
Copyright
Copyright © Cambridge University Press 1981

References

REFERENCES

[1]Borel, A.. Density properties for certain subgroups of semi-simple groups without compact factors. Annals of Math. 72 (1960), 179–188.CrossRefGoogle Scholar
[2]Borel, A. & Tits, J.. Groupes reductifs. Publ. Math. I.H.E.S. 27 (1965), 55–150.CrossRefGoogle Scholar
[3]Connes, A., Feldman, J. & Weiss, B.. To appear.Google Scholar
[4]Dye, H. A.. On groups of measure preserving transformations, I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
[5]Feldman, J. & Moore, C. C.. Ergodic equivalence relations, cohomology, and Von Neumann algebras, I. Trans. Amer. Math. Soc. 234 (1977), 289–324.CrossRefGoogle Scholar
[6]Feldman, J., Hahn, P. & Moore, C. C.. Orbit structure and countable sections for actions of continuous groups. Advances in Math. 28 (1978), 186–230.CrossRefGoogle Scholar
[7]Furstenberg, H.. Boundary theory and stochastic processes on homogeneous spaces. In Harmonic Analysis on Homogeneous Spaces, pp. 193–229. Symp. in Pure Math. Williamstown, Mass., 1972.Google Scholar
[8]Mackey, G. W.. Point realizations of transformation groups. III. J. Math. 6 (1962), 327–335.Google Scholar
[9]Mackey, G. W.. Ergodic theory and virtual groups. Math. Ann. 166 (1966), 187–207.CrossRefGoogle Scholar
[10]Margulis, G. A.. Non-uniform lattices in semisimple algebraic groups. In Lie Groups and Their Representations (ed. Gelfand, I. M.), pp. 371–553. Wiley: New York, 1975.Google Scholar
[11]Margulis, G. A.. Discrete groups of motions of manifolds of non-positive curvature. Amer. Math. Society Translations 109 (1977), 33–45.CrossRefGoogle Scholar
[12]Margulis, G. A.. Arithmeticity of irreducible lattices in semisimple groups of rank greater than 1. Appendix to Russian translation of Ragunathan, M.. Discrete Subgroups of Lie Groups. Mir: Moscow, 1977. (In Russian.)Google Scholar
[13]Moore, C. C.. Ergodicity of flows on homogeneous spaces. Amer. J. Math. 88 (1966), 154–178.CrossRefGoogle Scholar
[14]Mostow, G. D.. Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms. Publ. Math. I.H.E.S. 34 (1967), 53–104.CrossRefGoogle Scholar
[15]Mostow, G. D.. Strong Rigidity of Locally Symmetric Spaces. Annals of Math. Studies, no. 78. Princeton Univ. Press: Princeton, N.J., 1973.Google Scholar
[16]Ornstein, D. & Weiss, B.. To appear.Google Scholar
[17]Ramsay, A.. Virtual groups and group actions. Advances in Math. 6 (1971), 253–322.CrossRefGoogle Scholar
[18]Series, C.. The Rohlin tower theorem and hyperfiniteness for actions of continuous groups. Israel J. Math. 30 (1978), 99–112.CrossRefGoogle Scholar
[19]Series, C.. Foliations of polynomial growth are hyperfinite. Israel J. Math. 34 (1979), 245–258.CrossRefGoogle Scholar
[20]Zimmer, R. J.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal. 27 (1978), 350–372.CrossRefGoogle Scholar
[21]Zimmer, R. J.. Orbit spaces of unitary representations, ergodic theory, and simple Lie groups. Annals of Math. 106 (1977), 573–588.CrossRefGoogle Scholar
[22]Zimmer, R. J.. Induced and amenable ergodic actions of Lie groups. Ann. Sci. Ec. Norm. Sup. 11 (1978), 407–428.CrossRefGoogle Scholar
[23]Zimmer, R. J.. An algebraic group associated to an ergodic diffeomorphism. Comp. Math. 43 (1981), 59–69.Google Scholar
[24]Zimmer, R. J.. Strong rigidity for ergodic actions of semisimple Lie groups. Annals of Math. 112 (1980), 511–529.CrossRefGoogle Scholar
[25]Zimmer, R. J.. On the cohomology of ergodic actions of semisimple Lie groups and discrete subgroups. Amer. J. Math. (in the press).Google Scholar
[26]Zimmer, R. J.. On the Mostow rigidity theorem and measurable foliations by hyperbolic space. Preprint.Google Scholar