Standard results in descriptive set theory provide sufficient conditions for a Borel set P ⊆ℕ^ℕ×ℕ^ℕ to admit a Borel uniformization, namely, when P has "small" sections or "large" sections. We consider an invariant analogue of these results: Given a Borel equivalence relation E and an E-invariant Borel set P with "small" or "large" sections, does P admit an E-invariant Borel uniformization? For a given Borel equivalence relation E, we show that every E-invariant Borel set P with "small" or "large" sections admits an E-invariant Borel uniformization if and only if E is smooth. We also compute the definable complexity of counterexamples in the case where E is not smooth, using category, measure, and Ramsey-theoretic methods. We provide two new proofs of a dichotomy of Miller classifying the pairs (E, P) such that P admits an E-invariant uniformization, for a Borel equivalence relation E and a Borel E-invariant set P with countable sections. In the process, we prove an ℵ_0-dimensional (𝔾_0, ℍ_0) dichotomy, generalizing dichotomies of Miller and Lecomte. We also show that the set of pairs (E, P) such that P has "large" sections and admits an E-invariant Borel uniformization is Σ^1_2-complete; in particular, there is no analog of Miller's dichotomy for P with "large" sections. Finally, we consider a less strict notion of invariant uniformization, where we select a countable nonempty subset of each section instead of a single point.

Nadkarni's Theorem asserts that for a countable Borel equivalence relation (CBER) exactly one of the following holds: (1) It has an invariant Borel probability measure or (2) it admits a Borel compression, i.e., a Borel injection that maps each equivalence class to a proper subset of it. We prove in this paper an effective version of Nadkarni's Theorem, which shows that if a CBER is effectively Borel, then either alternative (1) above holds or else it admits an effectively Borel compression. As a consequence if a CBER is effectively Borel and admits a Borel compression, then it actually admits an effectively Borel compression. We also prove an effective version of the ergodic decomposition theorem. Finally a counterexample is given to show that alternative (1) above does not admit an effective version.

Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as $K_\sigma$ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness.

This paper is an introduction and survey of a “global” theory of measure preserving equivalence relations and graphs. In this theory one views a measure preserving equivalence relation or graph as a point in an appropriate topological space and then studies the properties of this space from a topological, descriptive set theoretic and dynamical point of view.

The class of ergodic, invariant probability Borel measure for the shift action of a countable group is a G(delta) set in the compact, metrizable space of probability Borel measures. We study in this paper the descriptive complexity of the class of ergodic, quasi-invariant probability Borel measures and show that for any infinite countable group Gamma it is Pi(0)(3)-hard, for the group Z it is Pi(0)(3)-complete, while for the free group F-infinity with infinite, countably many generators it is Pi(0)(alpha)-complete, for some ordinal a with 3 <= alpha <= omega + 2. The exact value of this ordinal is unknown.

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of $\mathbb{F}_2$ which are all invariant and weakly mixing with respect to the action of $\text{Aut}(\mathbb{F}_2)$. Moreover, for amenable groups $\Gamma\leq \Delta$, we obtain that the standard co-induction operation from the space of weak equivalence classes of $\Gamma$ to the space of weak equivalence classes of $\Delta$ is continuous if and only if $[\Delta :\Gamma]<\infty$ or $\text{core}_\Delta(\Gamma)$ is trivial. For general groups we obtain that the co-induction operation is not continuous when $[\Delta:\Gamma]=\infty$. This answers a question raised by Burton and Kechris. Independently such an answer was also obtained, using a different method, by Bernshteyn.

In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of F2 which are all invariant and weakly mixing with respect to the action of Aut(F2). Moreover, for amenable groups Γ ≤ ∆, we obtain that the standard co-induction operation from the space of weak equivalence classes of Γ to the space of weak equivalence classes of ∆ is continuous if and only if [∆ : Γ] < ∞ or core∆(Γ) is trivial. For general groups we obtain that the co-induction operation is not continuous when [∆ : Γ] = ∞. This answers a question raised by Burton and Kechris in [BK18]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [B18].

For a class $\mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there is a Borel way to put a structure in $\mathcal K$ on each $E$-equivalence class. We study in this paper the global structure of the classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$. We show that $\mathcal K$-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$, under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on $\mathcal K$-structurability of various model-theoretic properties of $\mathcal K$. In particular, we characterize the $\mathcal K$ such that every $\mathcal K$-structurable equivalence relation is smooth, answering a question of Marks.

We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound, which is not sharp because the relation is Borel.

We show that weak containment of free ergodic measure-preserving actions of $\mathbf{F}_\infty$ is not equivalent to weak containment of the corresponding Koopman representations. This result is based on the construction of an invariant random subgroup of $\mathbf{F}_\infty$ which is supported on the maximal actions.

We study in this paper ordered finite measure algebras from the point of view of Fraisse and Ramsey theory. We also propose an open problem, which is a homogeneous version of the Dual Ramsey Theorem of Graham-Rothschild, and derive consequences of a positive answer to the study of the topological dynamics of the automorphism group of a standard probability space and also the group of measure preserving homeomorphisms of the Cantor space.

It is shown that, within L(R), the smallest inner model of set theory containing the reals, the axiom of determinacy is equivalent to the existence of arbitrarily large cardinals below O with the strong partition property K -> (K)".