Alexander Kechris

• California Institute of Technology

This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible...

Standard results in descriptive set theory provide sufficient conditions for a set $P \subseteq \mathbb{N}^\mathbb{N} \times \mathbb{N}^\mathbb{N}$ 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 $E$-inva...

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 The...

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....

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....

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Large Cardinals, Determinacy and Other Topics is the final volume in a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussio...

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....

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...

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...

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...

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...

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 show that Tarski's concept of cardinal algebra appears naturally in the context of the current theory of Borel equivalence relations. As a result one can apply Tarski's theory to discover a number of interesting laws governing the structure of Borel equivalence relations, which, in retrospect rather surprisingly, have not been realized before.

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 va...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Ordinal Definability and Recursion Theory is the third in a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of resear...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Ordinal Definability and Recursion Theory is the third in a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of resear...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Ordinal Definability and Recursion Theory is the third in a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of resear...

A topological group G is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the profinite groups and (Qp, +) but our main interest here will be on non-locally compact groups. In recent years ther...

We establish the generic inexistence of stationary Borel probability measures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no σ-compact realization.

Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.

We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first e...

A topological group G is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the profinite groups and (ℚ_p, +) but our main interest here will be on non-locally compact groups. In recent years the...

It is shown that the translation action of the free group with n generators on its profinite completion is the maximum, in the sense of weak containment, measure preserving action of this group. Using also a result of Abért-Nikolov this is used to give a new proof of Gaboriau’s theorem that the cost of this group is equal to n. A similar maximality...

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU G is Borel inseparable fromU H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures for...

The twelfth Appalachian Set Theory workshop was held at Vanderbilt University in Nashville on October 30, 2010. The lecturer was Alexander S. Kechris. As a graduate student Robin D. Tucker-Drob assisted in writing this chapter, which is based on the workshop lectures. Dedicated to the memory of Greg Hjorth (1963–2011) The last two decades have seen...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Wadge Degrees and Projective Ordinals is the second of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Wadge Degrees and Projective Ordinals is the second of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Wadge Degrees and Projective Ordinals is the second of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Wadge Degrees and Projective Ordinals is the second of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research...

0 Introduction Let L be a countable first-order language. A class K of finite L-structures is called a Frassé class if it contains structures of arbitrarily large (finite) cardinality, is countable (in the sense that it contains only countably many isomorphism types) and satisfies the following: i) Hereditary property (HP): If B ∈ K and A can be em...

We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research de...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research de...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research de...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research de...

This paper was circulated in handwritten form in December 1980 and contained Sections 1-7 below. There are two additionalSections 8 and 9 here that contain further material and comments.

We study a positive-definite function associated to a measure-preserving equivalence relation on a standard probability space and use it to measure quantitatively the proximity of subequivalence relations. This is combined with a recent co-inducing construction of Epstein to produce new kinds of mixing actions of an arbitrary infinite discrete grou...

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.

The proceedings of the Los Angeles Caltech-UCLA ‘Cabal Seminar’ were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research de...

The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research de...

In this paper we develop a method for applying "forcing techniques" (or equivalently "category arguments") in the analytical hierarchy. We present here the basic general theory and some simple applications. In a subsequent publication, which is in preparation, further and technically more complicated applications will be given, including perfect se...

We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e....

In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on M^X, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the...

Without Abstract

It is the purpose of these notes to give an informal exposition of several recent results in Descriptive Set Theory, all centering around the notion of a scale. This was first isolated explicitly in the generalization of the Uniformization Theorem on the hypothesis of projective determinacy [14], but is surely implicit in some of the classical proo...

This exposition is a sequel to Kechris [1978]. Its main purpose is to show how set theoretical techniques, among them infinite exponent partition relations can be used to produce homogeneous trees for projective sets. The work here is again understood as being carried completely within L[R], with the hypothesis that AD+DC holds in this model.

This paper contains an exposition of certain aspects of the theory of Spector classes. The general plan is as follows: In Section 1 we present a review of the structure theory, mainly developed in Chapter 9 of Moschovakis [1]. In Section 2 we give a comprehensive list of examples from various areas of definability theory. Finally, in Section 3, we...

Working in the context of Projective Determinacy (PD), we introduce and study in this paper a countable ∏^1_(2n+1) set of reals Q_(2n+l) and an associated real y^0_(2n+l) for each n ≥ 0 (real means element of ω^ω in this paper). Our theory has analytical (descriptive set theoretic) as well as set theoretic aspects, strongly interrelated with each o...

Without Abstract

In this paper we present a simple general method for demonstrating that in certain function spaces various sets consisting of functions that exhibit at every point a prescribed kind of singularity form a coanalytic but not Borel set. We illustrate this method by providing new proofs that the set of nowhere differential continuous functions on [0,1]...

Assuming ZF + DC + AD Moschovakis (see [Ml]) has shown that if there is a surjection π : R → λ from the reals (R = ω^ω in this paper) onto an ordinal λ, then there is a surjection π^* : R → p(λ) from the reals onto the power set of λ. Let us denote by β(λ) the set of ultrafilters on λ. The question was raised whether there is an analog of Moschovak...

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Tru...

Contents. 14. Preliminaries 15. Countable Borel Equivalence Relations 16. More on Invariant Measures 17. Graphings of Equivalence Relations 18. Cost of an Equivalence Relation 19. Treeings of an Equivalence Relation 20. The Cost of a Smooth Equivalence Relation 21. The Cost of a Complete Section 22. Cost and Hyperfiniteness 23. Joins 24. Commuting...

5. Amenable Groups 6. Hyperfiniteness 7. Dye’s Theorem 8. Quasi-invariant Measures 9. Amenable Equivalence Relations 10. Amenability vs. Hyperfiniteness 11. Groups of Polynomial Growth 12. Generic Hyperfiniteness 13. Generic Compressibility

1. Group Actions and Equivalence Relations 2. Invariant Measures 3. Ergodicity 4. Isomorphism and Orbit Equivalence

Incluye bibliografía e índice

We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.

This volume comprises articles from four outstanding researchers who work at the cusp of analysis and logic. The emphasis is on active research topics; many results are presented that have not been published before and open problems are formulated. Considerable effort has been made by the authors to integrate their articles and make them accessible...

A b s t r a c t. We investigate some connections between the Frassé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable struc-tures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal f...

We study the classification problem of Polish metric spaces up to isometry and the isometry groups of Polish metric spaces. In the framework of the descriptive set theory of definable equivalence relations, we determine the exact complexity of various classification problems concerning Polish metric spaces. We start with the class of all Polish met...

Let G be a countable group and X a Borel G-space. Then it is clear that is a Borel equivalence relation and every one of its equivalence class is countable.

We cannot end before at least briefly discussing one other spectacular result of the 90's. We recall that the homogeneous Banach space problem (P5) is: If X is isomorphic to all Y ⊆ X, is X isomorphic to l2? This was solved by combining two beautiful pieces of work, Gowers' dichotomy theorem (Theorem 3.1) and the following theorem of Komorowski and...

In this chapter we prove isomorphism theorems for ultrapowers and ultraproducts of normed space structures. These results show that there is a very tight connection between (a) properties that are preserved under the ultraproduct construction and (b) properties that are expressible using the logic for normed space structures that is described in th...

In this chapter we introduce the key ingredients of the logic for normed space structures that is described in this paper. These are the positive bounded formulas and the concept of approximate satisfaction of such formulas in normed space structures. Let L be a signature for a normed space structure ℳ based on (M^(s) ∣ s ∈ S). Recall that S has a...