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Admissible subgroups of full ergodic groups

Published online by Cambridge University Press:  14 October 2010

Andrey Fedorov  and
Ben-Zion Rubshtein
Andrey Fedorov
Affiliation:
Astrakhan State Pedogogic Institute, Astrakhan, Russia
Ben-Zion Rubshtein
Affiliation:
Ben-Gurion University of the Negev, Beer-Sheva, Israel
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Abstract

Let G be a countable group of automorphisms of a Lebesgue space (X, m) and let [G] be the full group of G. For a pair of countable ergodic subgroups H1 and H2 of [G], the following problem is considered: when are the full subgroups [H1] and [H2] conjugate in the normalizer N[G] = {g ∈ Aut X: g[G]g-1 = [G]} of [G]. A complete solution of the problem is given in the case when [G] is an approximately finite group of type II and [H] is admissible, in the sense that there exists an ergodic subgroup [H0] of [G] and a countable subgroup Γ ⊂ N[H0] consisting of automorphisms which are outer for [H0], such that [H0] ⊂ [G] and the full subgroup [Ho, Γ] generated by [H0] and Γ coincides with [G].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1221 - 1239
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[BGl] Bezuglyi, S. I. and Golodets, V. Ja.. Outer conjugation of actions of countable groups on a measure space. Preprint. 2–84. FTINT. AN USSR. Kharkov.Google Scholar
[BG2] Bezuglyi, S. I. and Golodets, V. Ja.. Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizer of type III full group. J. Fund. Anal. 60 (1985), 341–369.CrossRefGoogle Scholar
[BG3] Bezuglyi, S. I. and Golodets, V. Ja.. Measure space transformations and outer conjugacy of countable amenable automorphism groups. Preprint. 28–85FTINT. AN USSR. Kharkov.Google Scholar
[BG4] Bezuglyi, S. I. and Golodets, V. Ja.. Izv. AN SSSR Math. Ser. 50 (1986), 643–660.Google Scholar
[CFW] Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1 (1981), 431–450.CrossRefGoogle Scholar
[CK] Connes, A. and Krieger, W.. Measure space automorphisms, the normalizers of their full groups and approximate finiteness. J. Fund. Anal. 24 (1977), 336–352.CrossRefGoogle Scholar
[D] Dye, H. A.. On groups of measure preserving transformations I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
[Fel] Fedorov, A. L.. Krieger's theorems for cocycles. Manuscript deposited in Izv. AN USSR no 1407 1985. 85-Dep. Febr 25. (in Russian).Google Scholar
[Fe2] Fedorov, A. L.. On weak closure of full groups of automorphisms Dokl. Akad. Nauk. USSR. 285 (1985), 49–52Google Scholar
[FM] Feldman, J. and Moore, C. C.. Ergodic equivalence relation, cohomology and von Neumann algebras I, II. Trans. AMS. 234 (1977), 289–324.CrossRefGoogle Scholar
[FSZ] Feldman, J., Sutherland, C. E. and Zimmer, R. J.. Subrelations of ergodic equivalence relations. Ergod. Th. & Dynam. Sys. 9 (1989), 239–269.CrossRefGoogle Scholar
[GS] Golodets, V. Ja. and Sinelchikov, S. I.. Existence and uniqueness of cocycles of ergodic automorphisms with dense ranges in amenable groups. Preprint. 19–83Kharkov 1983.Google Scholar
[HO] Hamachi, T. and Osikawa, M.. Ergodic groups of automorphisms and Krieger's theorems. Mem. math. Sci. Kelo Univ. 3 (1981), 1–113.Google Scholar
[K] Krieger, W.. On ergodic flows and isomorphism of factors. Math. Ann. 223 (1976), 19–70CrossRefGoogle Scholar
[Rol] Rokhlin, V. A.. Basic concepts of measure theory Math. Sb. 67 (1949), 107–150.Google Scholar
[Ro2] Rokhlin, V. A.. Lectures on entropy theory of transformations with invariant measure. Usp. Mat. Nauk. 12.5 (1967), 3–56.Google Scholar
[RuFel] Rubshtein, B. A. and Fedorov, A. L.. Subgroups of a full group 1. Manuscript deposited in UzNIITI no 185 Uz-D84. 1984, 1–39 (in Russian).Google Scholar
[RuFe2] Rubshtein, B. A. and Fedorov, A. L.. Subgroups of a full group II. Manuscript deposited in UzNIITI no 186 Uz-D84. 1984, 1–25 (in Russian).Google Scholar
[RuFe3] Rubshtein, B. A. and Fedorov, A. L.. On subgroups of a full ergodic approximately finite group of measure preserving automorphisms of a Lebesgue space. Func. Anal. 20 (1986), 82–83 (in English: 156-158).Google Scholar
[VFe] Vershik, A. M. and Fedorov, A. L.. Trajectory theory. Modern Problems in Math. 26 (1985), 173–212. (In English: J. Sov. Math. 38 (1987), 1789-1822).Google Scholar
[VekFe] Veksler, A. S. and Fedorov, A. L.. On conjugacy of homomorphisms of locally compact groups into outer groups of measured equivalence relations. Fund. Anal. Appl. 22 (1988), 74–75.Google Scholar
[ViRuFel] Vinokurov, V. G., Rubshtein, B. A. and Fedorov, A. L.. Lebesgue Spaces and their Measurable Partitions. Tashkent Univ. 1986 76 pp.Google Scholar
[ViRuFe2] Vinokurov, V. G., Rubshtein, B. A. and Fedorov, A. L.. Normal subgroupoids and extensions of measured groupoids. Russ. Math. Survey 43 (1988), 231–232.CrossRefGoogle Scholar
[Z] Zimmer, R.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Fund. Anal. 27 (1978), 350–372.CrossRefGoogle Scholar