Topology and convexity in the space of actions modulo weak equivalence

Burton, Paul

doi:10.1017/etds.2016.133

Cited by 4 publications

(20 citation statements)

References 11 publications

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“…In [T-D1, Theorem 5.1] it is shown that if we transfer the relative topology of this subset back to A ( , X, µ) via this bijection, we obtain a compact, metrizable topology on A ( , X, µ). It was shown in [Bu,Theorem 3.1] that this coincides with the topology defined before.…”

supporting

confidence: 72%

“…THEOREM 9.3. [Bu,Proposition 5.1]; see also [T-D1, Theorem 1.8], and [E,Theorem 9] If is amenable, then a b ⇐⇒ type(a) = type(b).…”

Section: Invariant Random Subgroupsmentioning

confidence: 99%

“…This is clear for amenable groups, since an amenable group has exactly one free stable weak equivalence class, namely that of the shift. If a group is not amenable, then it has at least two free stable weak equivalence classes, and then it follows that there must exist a non-ergodic one, using [Bu,Proposition 7.3] (see also Theorem 11.29 below).…”

Section: Stable Weak Containmentmentioning

confidence: 99%

“…THEOREM 11.25. [Bu,Corollary 4.1] The space A ( , X, µ) is path connected. Similarly for the subspaces A θ ( , X, µ) and, in particular, FR ( , X, µ).…”

Section: (I)mentioning

confidence: 99%

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Weak containment of measure-preserving group actions

Burton¹,

Kechris²

2019

Ergod. Th. Dynam. Sys.

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IntroductionThis paper is a contribution to the study of the global structure of measure preserving actions of countable (discrete) groups on non-atomic standard probability spaces. For such a group Γ and space (X, µ), we let A(Γ, X, µ) be the space of measure preserving actions of Γ on (X, µ). In the book [K] a hierarchical notion of complexity of such actions, called weak containment, was introduced, motivated by analogous notions of weak containment of unitary representations. Roughly speaking an action a ∈ A(Γ, X, µ) is weakly contained in an action b ∈ A(Γ, X, µ), in symbols a b, if the action of a on finitely many group elements and finitely many Borel sets in X can be simulated with arbitrarily small error by the action of b. We also denote by a ≃ b ⇐⇒ a b & b a the corresponding notion of weak equivalence.This notion of equivalence is much coarser than the notion of isomorphism (conjugacy) a ∼ = b. 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 A(Γ, X, µ)/ ∼ = of actions modulo isomorphism. On the other hand, weak equivalence is a smooth equivalence relation and the space of weak equivalence classes A(Γ, X, µ)/ ≃ 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, combinatorial parameters, etc., turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to 1 the pre-order of weak containment, a fact which 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. We include detailed references to the literature, where the reader can find proofs of theorems that are discussed here. We do include a few proofs either of results that have not appeared before or when we thought that a more detailed or sometimes a simplified presentation is desirable.The paper is organized as follows. Section 1 reviews concepts of weak containment for unitary representations. In Section 2, we define weak containment for actions and provide several equivalent reformulations. In Section 3, we start the study of the pre-order of weak containment and discuss its relationship with concepts such as freeness, ergodicity, strong ergodicity, and co-induction for group actions. In Section 4, we discuss the connection of weak containment of actions with that of their Koopman representations. Section 5 continues the study of the pre-order of weak containment, concentrating on the existence and description of a maximum action. In Section 6, we discuss the relationship of weak containment to the classical concept of factoring, which is a strong form of weak containment, including some recently established rigidity phenomena concerning these notions. Section 7 surveys the invariance and monotonicity properties of various pa...

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supporting

confidence: 72%

“…THEOREM 9.3. [Bu,Proposition 5.1]; see also [T-D1, Theorem 1.8], and [E,Theorem 9] If is amenable, then a b ⇐⇒ type(a) = type(b).…”

Section: Invariant Random Subgroupsmentioning

confidence: 99%

Section: Stable Weak Containmentmentioning

confidence: 99%

“…THEOREM 11.25. [Bu,Corollary 4.1] The space A ( , X, µ) is path connected. Similarly for the subspaces A θ ( , X, µ) and, in particular, FR ( , X, µ).…”

Section: (I)mentioning

confidence: 99%

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Weak containment of measure-preserving group actions

Burton¹,

Kechris²

2019

Ergod. Th. Dynam. Sys.

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“…This paper also establishes equivalent definitions of weak containment in terms of the space of all actions and ultraproducts of actions. this result inspired us to finish this work[Bur15]. So thanks, Peter.…”

mentioning

confidence: 77%

The space of stable weak equivalence classes of measure-preserving actions

Bowen¹,

Tucker-Drob²

2018

Journal of Functional Analysis

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The concept of (stable) weak containment for measure-preserving actions of a countable group Γ is analogous to the classical notion of (stable) weak containment of unitary representations. If Γ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Γ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when Γ = Z this simplex has only a countable set of extreme points but when Γ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Γ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points.

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Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem

Bernshteyn

2019

Isr. J. Math.

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A b s t r ac t . Let Γ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action Γ ñ pX, µq and a map f P L 1 pX, µq, and to compare the global average ş f dµ of f to the pointwise averages |D|´1 ř δPD f pδ¨xq, where x P X and D is a nonempty finite subset of Γ. The basic hope is that, when D runs over a suitably chosen infinite sequence, these pointwise averages should converge to the global value for µ-almost all x.In this paper we prove several results that refine the above basic paradigm by uniformly controlling the averages over specific sets D rather than considering their limit as |D| Ñ 8. Our results include ergodic theorems for the Bernoulli shift action Γ ñ pr0; 1s Γ , λ Γ q and strengthenings of the theorem of Abért and Weiss that the shift is weakly contained in every free p.m.p. action of Γ. In particular, we establish a purely Borel version of the Abért-Weiss theorem for finitely generated groups of subexponential growth. The central role in our arguments is played by the recently introduced measurable versions of the Lovász Local Lemma, due to the current author and to Csóka, Grabowski, Máthé, Pikhurko, and Tyros.versions of the LLL can provide a µ-measurable (or sometimes even Borel) function f : X Ñ C that satisfies the constraints B, or at least does so on a "large" subset of X. Such results appear to be particularly relevant in ergodic theory, since many concepts pertaining to measure-preserving group actions are phrased in terms of measurable partitions of the underlying probability space-which can naturally be thought of as measurable colorings. Some ergodic-theoretic applications of the LLL can be found in [Ber16; Ber18]. Here we present further consequences of the LLL in measurable dynamics, specifically in the study of ergodic averages and of weak containment of measure-preserving group actions.Our arguments employ a general approach that is standard in combinatorics, in particular in graph coloring theory (see, e.g., the book [MR02] for many examples). The first step is to use concentration of measure to obtain strong upper bounds on probabilities of certain "bad" random events; the LLL is then invoked to eliminate all the "bad" events. Nontrivial results can also be derived by combining the concentration of measure bounds with more classical tools, such as the Borel-Cantelli lemma (Theorem 2.1 below is as an example). Roughly speaking, using the LLL instead of the Borel-Cantelli lemma results in replacing pointwise convergence with approximation in the 8-norm.Acknowledgement. I am very grateful to Anush Tserunyan for many insightful discussions and to the anonymous referee for helpful suggestions. . S tat e m e n t s o f r e s u lt sThroughout, Γ denotes a countably infinite group with identity element 1. We study probability measure-preserving (p.m.p.) actions of Γ, i.e., actions of the form α : Γ ñ pX, µq, where pX, µq is a standard probability space and the measure µ is α-invariant. We also consider, more generall...

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Topology and convexity in the space of actions modulo weak equivalence

Cited by 4 publications

References 11 publications

Weak containment of measure-preserving group actions

Weak containment of measure-preserving group actions

The space of stable weak equivalence classes of measure-preserving actions

Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem

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