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Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel… (More)
An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π 1 1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr's Theorem. In the new setting, while K-trivial sets exist that are not hyper-arithmetical, each low for random set is. Finally we study a very… (More)
An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π 1 1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr's Theorem. In the new setting, while K-trivial sets exist that are not hyperarithmetical, each low for random set is. Finally, we begin to study… (More)
Using recent work on the algebraic structure of topological full groups of minimal subshifts, we prove that the isomorphism relation on the space of infinite finitely generated simple amenable groups is not smooth. As an application, we deduce that there does not exist an isomorphism-invariant Borel map which selects a just-infinite quotient of each… (More)
Assuming AD + DC(R), we characterize the self-dual boldface point-classes which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank ≤ ω ξ 1 , and those of Wadge rank < ω ξ 1 when ξ is limit.
For any countable Borel equivalence relation E on a standard Borel space X, there is a Borel function θ from X to the 2-generated groups such that xEy ⇔ θ(x) ∼ = θ(y) .