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This volume provides a self-contained introduction to some topics
in orbit equivalence theory, a branch of ergodic theory. The first
two chapters focus on hyperfiniteness and amenability. Included
here are proofs of Dye's theorem that probability
measure-preserving, ergodic actions of the integers are orbit
equivalent and of the theorem of Connes-Feldman-Weiss identifying
amenability and hyperfiniteness for non-singular equivalence
relations. The presentation here is often influenced by descriptive
set theory, and Borel and generic analogs of various results are
discussed. The final chapter is a detailed account of Gaboriau's
recent results on the theory of costs for equivalence relations and
groups and its applications to proving rigidity theorems for
actions of free groups.
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