# On subrelations of ergodic measured type III equivalence relations

Colloquium Mathematicae (2000)

- Volume: 84/85, Issue: 1, page 13-22
- ISSN: 0010-1354

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topDanilenko, Alexandre. "On subrelations of ergodic measured type III equivalence relations." Colloquium Mathematicae 84/85.1 (2000): 13-22. <http://eudml.org/doc/210792>.

@article{Danilenko2000,

abstract = {We discuss the classification up to orbit equivalence of inclusions 𝑆 ⊂ ℛ of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of 𝑆 ⊂ ℛ), an ergodic nonsingular ℝ-flow V and a homomorphism of G to the centralizer of V.},

author = {Danilenko, Alexandre},

journal = {Colloquium Mathematicae},

keywords = {orbit theory; ergodic relations},

language = {eng},

number = {1},

pages = {13-22},

title = {On subrelations of ergodic measured type III equivalence relations},

url = {http://eudml.org/doc/210792},

volume = {84/85},

year = {2000},

}

TY - JOUR

AU - Danilenko, Alexandre

TI - On subrelations of ergodic measured type III equivalence relations

JO - Colloquium Mathematicae

PY - 2000

VL - 84/85

IS - 1

SP - 13

EP - 22

AB - We discuss the classification up to orbit equivalence of inclusions 𝑆 ⊂ ℛ of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of 𝑆 ⊂ ℛ), an ergodic nonsingular ℝ-flow V and a homomorphism of G to the centralizer of V.

LA - eng

KW - orbit theory; ergodic relations

UR - http://eudml.org/doc/210792

ER -

## References

top- [Da1] A. I. Danilenko, Comparison of cocycles of measured equivalence relations and lifting problems, Ergodic Theory Dynam. Systems 18 (1998), 125-151. Zbl0919.28015
- [Da2] A. I. Danilenko, Quasinormal subrelations of ergodic equivalence relations, Proc. Amer. Math. Soc. 126 (1998), 3361-3370. Zbl0917.28019
- [Dy] H. Dye, On groups of measure preserving transformations. I, Amer. J. Math. 81 (1959), 119-159. Zbl0087.11501
- [FM] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras I, Trans. Amer. Math. Soc. 234 (1977), 289-324. Zbl0369.22009
- [FSZ] J. Feldman, C. Sutherland and R. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynam. Systems 9 (1989), 239-269. Zbl0654.22003
- [Ge] M. Gerber, Factor orbit equivalence and classification of finite extensions of ergodic transformations, Israel J. Math. 57 (1987), 28-48. Zbl0626.28012
- [GS1] V. Ya. Golodets and S. D. Sinel'shchikov, Amenable ergodic actions of groups and images of cocycles, Soviet Math. Dokl. 41 (1990), 523-525.
- [GS2] V. Ya. Golodets and S. D. Sinel'shchikov, Classification and structure of cocycles of amenable ergodic equivalence relations, J. Funct. Anal. 121 (1994), 455-485. Zbl0821.28010
- [Ha] T. Hamachi, Suborbits and group extensions of flows, Israel J. Math. 100 (1997), 249-283. Zbl0880.46046
- [HO] T. Hamachi and M. Osikawa, Ergodic groups of automorphisms and Krieger's theorems, Sem. Math. Sci. Keio Univ. 3 (1981). Zbl0472.28015
- [Kr] W. Krieger, On ergodic flows and isomorphism of factors, Math. Ann. 223 (1976), 19-70. Zbl0332.46045
- [Sc] K. Schmidt, On recurrence, Z. Wahrsch. Verw. Gebiete 68 (1984), 75-95. Zbl0535.60003
- [Zi] R. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409. Zbl0334.28015