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Cocycle superrigidity for profinite actions of property (T) groups. (English) Zbl 1235.37005

The main result of the paper is Theorem A on orbit equivalence (OE) superrigidity. As a consequence of this result, the author constructs uncountably many non-OE profinite actions for the arithmetic groups \(SL_n (\mathbb Z)\) (\(n \geq 3\)) and their finite subgroups, as well as for the groups \(SL_m (\mathbb Z) \ltimes{\mathbb Z}^m\) (\(m \geq 2\)). The author deduces Theorem A as a consequence of a theorem (Theorem B) on cocycle superrigidity.
“Let \(\Gamma \curvearrowright X\) be a free ergodic measure-preserving profinite action (i.e., an inverse limit of actions \(\Gamma \curvearrowright X_n\) with \(X_n\) finite) of a countable property (T) group \(\Gamma \) (more generally, of a group \(\Gamma\) which admits an infinite normal subgroup \(\Gamma_0\) such that the inclusion \(\Gamma_0 \subset \Gamma\) has relative property (T) and \({\Gamma} \diagup {\Gamma_0}\) is finitely generated) on a standard probability space \(X\). The author proves that if \(\omega:\Gamma \times X \to \Lambda\) is a measurable cocycle with values in a countable group \(\Lambda\), then \(\omega\) is a cohomologous to a cocycle \(\omega'\) which factors through the map \(\Gamma \times X \to \Gamma \times X_n\), for some \(n\). As a corollary, he shows that any free ergodic measure-preserving action \(\Lambda \curvearrowright Y\) comes from a (virtual) conjugacy of actions.”
The notion of property (T) for locally compact groups was defined by D. A. Kazhdan [Funct. Anal. Appl. 1, 63–65 (1967); translation from Funkts. Anal. Prilozh. 1, No. 1, 71–74 (1967; Zbl 0168.27602)] and the notion of relative property (T) for inclusion of countable groups \(\Gamma_0 \subset \Gamma\) was defined by G. A. Margulis [Ergodic Theory Dyn. Syst. 2, 383–396 (1982; Zbl 0532.28012)].
The concept of superrigidity was introduced by G. D. Mostow [Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies. No. 78. Princeton, NJ: Princeton University Press and University of Tokyo Press (1973; Zbl 0265.53039)] and by G. A. Margulis [Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 17. Berlin: Springer (1991; Zbl 0732.22008)] in the context of studying the structure of lattices in rank one and higher rank Lie groups, respectively.
The paper under review presents a new class of orbit equivalent superrigid actions. Previously, the first result of this type was obtained by A. Furman [Ann. Math. (2) 150, No. 3, 1059–1081 (1999; Zbl 0943.22013); ibid. 150, No. 3, 1083–1108 (1999; Zbl 0943.22012)], who combined the cocycle superrigidity by R. J. Zimmer [Ergodic theory and semisimple groups. Monographs in Mathematics, Vol. 81. Boston-Basel-Stuttgart: Birkhäuser (1984; Zbl 0571.58015)] with ideas from geometric group theory to show that the actions \(SL_{n} (\mathbb Z) \curvearrowright T^n (n \geq 3)\) are OE superrigid. The deformable actions of rigid groups are OE superrigid by [S. Popa, in: Proceedings of the international congress of mathematicians (ICM) 2006. Volume I: Plenary lectures and ceremonies. Zürich: European Mathematical Society (EMS). 445–477 (2007; Zbl 1132.46038)].
A detailed analysis of several applications illustrates very well the most important points of the author’s approach.

MSC:

37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
28D15 General groups of measure-preserving transformations
46L36 Classification of factors
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References:

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