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Weak and approximate equivalence of group actions in the framework of ultrapower Cartan inclusions


The purpose of this dissertation is to study some new notions of equivalence

of measure preserving group actions on probability spaces. These

include approximate versions of conjugacy and orbit equivalence (OE) as

well as a notion of weak OE, which is engendered by the notion of weak

conjugacy that was recently introduced by A. Kechris. We study the relationship

between our new notions and the classical ones. For instance,

we show that approximate conjugacy is the same as conjugacy for actions

of groups with Kazhdan's property (T) whereas for groups with infinite

amenable quotients these notions are very much distinct. We also use

results of Monod-Shalom, Kida, and Chifan-Kida to deduce superrigidity

results within the paradigm of approximate conjugacy and OE. Moreover,

we show that a number of invariants and properties are preserved by weak

OE, including strong ergodicity. This allows us to deduce that any nonamenable

group without Kazhdan's property (T) has at least two weakly

orbit inequivalent actions. This dissertation is largely based on the author's

joint work with Professor Sorin Popa in the paper [AP15].

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