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Trees and amenable equivalence relations

Published online by Cambridge University Press:  19 September 2008

Scot Adams
Scot Adams
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305, USA
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Abstract

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Let R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 1 , March 1990 , pp. 1 - 14
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Arveson, W.. An Invitation to C*-algebras. New York: Springer-Verlag, 1976.Google Scholar
[2]Connes, A., Feldman, J. & Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. Dynam. Sys. 1 (1981), 431–450.Google Scholar
[3]Feldman, J., Hahn, P. & Moore, C.. Orbit structure and countable sections for actions of continuous groups. Adv. Math. 28 (1978), 186–230.Google Scholar
[4]Feldman, J. & Moore, C.. Ergodic equivalence relations, cohomology and von Neumann algebras I. Trans. Amer. Math. Soc. 234 (1977), 289–324.CrossRefGoogle Scholar
[5]Feldman, J. & Moore, C.. Ergodic equivalence relations, cohomology and von Neumann algebras II. Trans. Amer. Math. Soc. 234 (1977), 325–359.CrossRefGoogle Scholar
[6]Mackey, G.. Ergodic theory and virtual groups. Math. Annalen 166 (1966), 187–207.Google Scholar
[7]Ramsay, A.. Virtual groups and group actions. Adv. Math. 6 (1971), 253–322.CrossRefGoogle Scholar
[8]Serre, J.-P.. Trees. New York: Springer-Verlag, 1980.Google Scholar
[9]Zimmer, R.. Extensions of ergodic group actions. Ill. J. Math. 20 (1976), 373–409.Google Scholar
[10]Zimmer, R.. Hyperfinite factors and amenable ergodic actions. Inv. Math. 41 (1977), 23–31.CrossRefGoogle Scholar
[11]Zimmer, R.. Orbit equivalence and rigidity of ergodic actions of Lie groups. Ergod. Th. & Dynam. Sys. 1 (1981), 237–253.Google Scholar
[12]Zimmer, R.. Curvature of leaves in amenable foliations. Amer. J. Math. 105 (1983), 1011–1022.Google Scholar
[13]Zimmer, R.. Ergodic Theory and Semisimple Groups. Boston: Birkhauser, 1984.Google Scholar