Invariant random subgroups and action versus representation maximality

Burton, Paul; Kechris, Alexander S.

doi:10.1090/proc/13591

Cited by 3 publications

(1 citation statement)

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“…ergodic theory [2,1,54,17,56,19], as well as many other subjects. While not quite the same object, we think that G-invariant random subgroups in the algebraic action situation will be important to the study of entropy theory for the same reasons that invariant random subgroups are of relevance to several fields of mathematics.…”

Section: Introductionmentioning

confidence: 99%

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

Hayes¹

2018

Preprint

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We continue our study of when topological and measure-theoretic entropy agree for algebraic action of sofic groups. Specifically, we provide a new abstract method to prove that an algebraic action is strongly sofic. The method is based on passing to a "limiting object" for sequences of measures which are asymptotically supported on topological microstates. This limiting object has a natural poset structure that we are able to exploit to prove a max-min principle: if the sofic approximation has ergodic centralizer, then the largest subgroup on which the action is a local weak * -limit of measures supported on topological microstates is equal to the smallest subgroup which absorbs all topological microstates. We are able to provide a version for the case when the centralizer is not ergodic. We give many applications, including show that for residually finite groups completely positive (lower) topological entropy (in the presence) is equivalent to completely positive (lower) measure entropy in the presence.

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Section: Introductionmentioning

confidence: 99%

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

Hayes¹

2018

Preprint

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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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Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy

Hayes¹

2019

Preprint

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We prove a max-min theorem for weak containment in the context of algebraic actions. Namely, we show that given an algebraic action G X, there is a maximal, closed G-invariant subgroup Y of X so that G (Y, m Y ) is weakly contained in a Bernoulli shift. This subgroup is also the minimal closed subgroup so that any action weakly contained in a Bernoulli shift is G X/Y -ergodic "in the presence of G X". We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal algebraic actions whose associated convolution operator is injective.We also deduce from our techniques that algebraic actions whose square summable homoclinic group is dense have completely positive entropy when the acting group is sofic.

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Invariant random subgroups and action versus representation maximality

Cited by 3 publications

References 8 publications

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

Local weak$^{*}$-Convergence, algebraic actions, and a max-min principle

Weak containment of measure-preserving group actions

Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy

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