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Subrelations of ergodic equivalence relations†

Published online by Cambridge University Press:  19 September 2008

J. Feldman ,
C. E. Sutherland  and
R. J. Zimmer
J. Feldman
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720, USA
C. E. Sutherland
Affiliation:
Mathematics Department, University of New South Wales, Kensington, NSW, 2033, Australia
R. J. Zimmer
Affiliation:
Mathematics Department, University of Chicago, Chicago, Illinois 60637, USA
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Abstract

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We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a group Q which serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) of Q on it; in the case where Q is amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 2 , June 1989 , pp. 239 - 269
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Brown, R. G.. Fibrations of groupoids. Journal of Algebra 15 (1) (1970), 103–132.CrossRefGoogle Scholar
[2]Chevalley, C.. Theory of Lie Groups (Princeton University Press: Princeton, 1946).Google Scholar
[3]Connes, A., Feldman, J. & Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. I (1981), 431–450.CrossRefGoogle Scholar
[4]Connes, A. & Jones, V.. Property T for von Neumann Algebras. Bull. Lon. Math. Soc. 17 (1985), 57–62.CrossRefGoogle Scholar
[5]Dye, H. A.. On groups of measure preserving transformations I and II. Amer. J. Math. 81 (1959), 119–159Google Scholar
and Amer. J. Math. 85 (1963), 551–576.Google Scholar
[6]Eberlein, P.. Geodesic flows on negatively curved manifolds I, II. Ann. Math. 95 (1972), 492–510Google Scholar
and T.A.M.S. 1978 (1973), 57–82.Google Scholar
[7]Eberlein, P.. Lattices in spaces of non-positive curvature. Ann. Math. III (1980), 455–467.Google Scholar
[8]Effros, E.. Transformation groups and C*-algebras. Ann. Math. 81 (1965), 38–55.Google Scholar
[9]Fabec, R. C.. Normal Ergodic actions and Extensions. Israel J. Math. 40 (2) (1981), 175–186.Google Scholar
[10]Feldman, J. & Moore, C. C.. Ergodic equivalence relations, cohomology and von Neumann algebras I and II. T.A.M.S. 234 (1977) 289–324 and 325–359.Google Scholar
[11]Gerber, M.. Classifying the finite extensions of an ergodic transformation up to factor orbit equivalence. Preprint.Google Scholar
[12]Golodets, V. Ya. & Sinelchikov, S. I.. Existence and uniqueness of cocycles of an ergodic automorphism with dense ranges in amenable groups. Preprint 19–83, FTINT AN USSR, Kharkov, 1–21.Google Scholar
[13]Jones, V. F. R.. Index for subfactors. Inventiones Math. 72 (1983), 1–25.CrossRefGoogle Scholar
[14]Mackey, G. W.. Borel spaces for groups and their duals. T.A.M.S. 85 (1957), 134–165.CrossRefGoogle Scholar
[15]Mackey, G. W.. Infinite dimensional group representations. B.A.M.S. 69 (1963), 628–686.Google Scholar
[16]Margulis, G. A.. Finiteness of quotient groups of discrete subgroups. Funct. Anal. Appl. 13 (1979), 178–187.CrossRefGoogle Scholar
[17]Ocneanu, A.. Actions of discrete amenable groups on von Neumann algebras. Springer Lecture Notes in Math. 1138 (Springer-Verlag: New York, 1985).Google Scholar
[18]Ornstein, D. & Weiss, B.. Ergodic theory of amenable group actions. Preprint.Google Scholar
[19]Ramsay, A.. Virtual groups and group actions. Adv. Math. 6 (1971), 253–322.CrossRefGoogle Scholar
[20]Schmidt, K.. Strong ergodicity and quotients of equivalence relations. To appear in Proc. of the Centre for Mathematical Analysis (Australian National University, Canberra).Google Scholar
[21]Sutherland, C. E.. Sub-equivalence relations and subalgebras of von Neumann algebras. In preparation.Google Scholar
[22]Sutherland, C. E.. A Borel parametrization of Polish groups. R.I.M.S. Kyoto University 21 (6) (1985), 1067–1086.CrossRefGoogle Scholar
[23]Varadarajan, V. S.. The Geometry of Quantum Theory. Vol II (van Nostrand Reinhold, New York, 1970).Google Scholar
[24]Zimmer, R. J.. Ergodic Theory and Semi-simple groups (Birkhauser, 1984).Google Scholar
[25]Zimmer, R. J.. Extensions of ergodic group actions. Illinois J. Math. 20 (1976), 373–409.CrossRefGoogle Scholar
[26]Zimmer, R. J.. Normal ergodic actions. J.F.A. 25 (3) (1977), 281–305.Google Scholar
[27]Zimmer, R. J.. Ergodic Theory, Semi-simple Groups and foliations by manifolds of negative curvature. Pub. Math. I.H.E.S. 55 (1982), 37–61.Google Scholar
[28]Bezuglyi, S. I. & Golodets, V. Ya.. Outer conjugation of actions of countable groups on a measure space. Preprint 2–84, F.T.I.N.T. AN U.S.S.R., Kharkov.Google Scholar
[29]Bezuglyi, S. I. & Golodets, V. Ya.. Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups. J.F.A. 60 (1985), 341–369.Google Scholar
[30]Bezuglyi, S. I. & Golodets, V. Ya.. Measure space transformations and outer conjugacy of countableamenable automorphism groups. Preprint 28–85, F.T.I.N.T. AN, U.S.S.R., Kharkov.Google Scholar