Contents Chapter 0. Preliminaries 0A. OB. OC. OD. OE. OF. Actions Equivalence relations Borel notions Measures Borel actions and measures Amenability Introduction 1 Historical remarks and overview of the main results. 9 9 9 10 11 11 12 Chapter 1. Actions of Free Groups and Treeable Equivalence Relations 15 Chapter 2. A Cocycle Reduction Result 21 The ad hoc notion of near hyperbolic. The main technical device under the assumption of i?o-ergodicity (2.2). Chapter 3. Some Applications 25 Various applications are presented for the structure of countable Borel equiv- alence relations under Borel reducibility. 3A. An "elementary" proof of existence of incomparables 25 3B. Further "elementary" proofs of theorems of Adams-Kechris 27 3C. "Elementary" proofs of results of Adams and Thomas 28 3D. Relative ergodicity and rigidity results for product group actions 37 Chapter 4. Factoring Homomorphisms 41 A further technical theorem without the assumption of £t)-ergodicity (4.2). A theorem on factoring homomorphisms of equivalence relations. Chapter 5. Further Applications 45 5A. Rigidity results for reducibility and stable orbit equivalence 45 5B. Products of hyperbolic groups 46 Chapter 6. Product Actions, I 49 Non-reducibility of mixing product group actions to products of treeable equivalence relations. Chapter 7. Product Actions, II 55 A parallel non-reducibility result for product actions of Ti x T2, where I \ has large approximate ergodic dimension and T2 is non-amenable, to smaller products of treeable equivalence relations.
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