Topics in Orbit Equivalence

1. Front Matter

   Pages N2-X

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2. I. Orbit Equivalence

   • Alexander S. Kechris, Benjamin D. Miller

   Pages 1-6

3. II. Amenability and Hyperfiniteness

   • Alexander S. Kechris, Benjamin D. Miller

   Pages 7-53

4. III. Costs of Equivalence Relations and Groups

   • Alexander S. Kechris, Benjamin D. Miller

   Pages 55-128

5. References

   • Alexander S. Kechris, Benjamin D. Miller

   Pages 129-130

6. Index

   • Alexander S. Kechris, Benjamin D. Miller

   Pages 131-134

7. Back Matter

   Pages 129-135

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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.

• Equivalence
• amenability
• cost
• equivalence relations
• hyperfiniteness
• orbit equivalence
• set
• set theory

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