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Orbit Equivalence and Measured Group Theory
 INTERNATIONAL CONGRESS OF MATHEMATICIANS (ICM), HYDERABAD: INDIA
, 2010
"... We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions. ..."
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Cited by 20 (0 self)
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We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.
Amenable actions and almost invariant sets
 Proc. Amer. Math. Soc
"... Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX, where M is a measure space. In parti ..."
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Abstract. In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ ↪→MX has almost invariant sets. 1.
Orbit equivalence rigidity and certain generalized Bernoulli shifts of the mapping class group, preprint
"... Abstract. We prove that any ergodic standard action (that is, an essentially free, measurepreserving action on a standard Borel space with a probability measure) of the mapping class group of a compact orientable surface with higher complexity has strong orbital rigidity: if there are ergodic stan ..."
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Abstract. We prove that any ergodic standard action (that is, an essentially free, measurepreserving action on a standard Borel space with a probability measure) of the mapping class group of a compact orientable surface with higher complexity has strong orbital rigidity: if there are ergodic standard actions of the mapping class group and a discrete group which are weakly orbit equivalent, then the two actions are virtually isomorphic. Moreover, we classify certain generalized Bernoulli shifts of the mapping class group up to orbit equivalence. This gives uncountably many examples of ergodic standard actions of the mapping class group which are mutually nonorbit equivalent. We prove similar statements for a finite direct product of mapping class groups.