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Bernoulli actions are weakly contained in any free action

Published online by Cambridge University Press:  07 February 2012

MIKLÓS ABÉRT  and
BENJAMIN WEISS
MIKLÓS ABÉRT
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, H-1053, Budapest, Hungary (email: [email protected])
BENJAMIN WEISS
Affiliation:
Hebrew University of Jerusalem, Institute of Mathematics, Jerusalem 91904, Israel (email: [email protected])
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Abstract

Let Γ be a countable group and let f be a free probability measure-preserving action of Γ. We show that all Bernoulli actions of Γ are weakly contained in f. It follows that for a finitely generated group Γ, the cost is maximal on Bernoulli actions for Γ and that all free factors of i.i.d. (independent and identically distributed) actions of Γ have the same cost. We also show that if f is ergodic, but not strongly ergodic, then f is weakly equivalent to f×I, where Idenotes the trivial action of Γ on the unit interval. This leads to a relative version of the Glasner–Weiss dichotomy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 323 - 333
Copyright
©2012 Cambridge University Press

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References

[1]Abért, M. and Nikolov, N.. Rank gradient, cost of groups and the rank versus Heegard genus problem, J. Eur. Math. Soc. to appear.Google Scholar
[2]Connes, A. and Weiss, B.. Property T and asymptotically invariant sequences. Israel J. Math 37 (1980), 209–210.Google Scholar
[3]Gaboriau, D.. Coût des relations d’équivalence et des groupes [Cost of equivalence relations and of groups]. Invent. Math. 139(1) (2000), 41–98 (in French).Google Scholar
[4]Glasner, E., Thouvenot, J.-P. and Weiss, B.. Every countable group has the weak Rohlin property. Bull. Lond. Math. Soc. 38 (2006), 932–936.Google Scholar
[5]Glasner, E. and Weiss, B.. Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7(5) (1997), 917–935.Google Scholar
[6]Jones, V. and Schmidt, K.. Asymptotically invariant sequences and approximate finiteness. Amer. J. Math. 109 (1987), 91–114.Google Scholar
[7]Kechris, A.. Global Aspects of Ergodic Group Actions (Mathematical Surveys and Monographs of the AMS, 160). American Mathematical Society, Providence, RI, 2010.Google Scholar
[8]Levitt, G.. On the cost of generating an equivalence relation. Ergod. Th. & Dynam. Sys. 15(6) (1995), 1173–1181.CrossRefGoogle Scholar
[9]Schmidt, K.. Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group-actions. Ergod. Th. & Dynam. Sys. 1(2) (1981), 223–236.CrossRefGoogle Scholar
[10]Schmidt, K.. Asymptotically invariant sequences and an action of SL(2,Z) on the 2-sphere. Israel J. Math. 37 (1980), 193–208.Google Scholar