Weak equivalence and non-classifiability of measure preserving actions

Tucker-Drob, Robin

doi:10.1017/etds.2013.40

Cited by 20 publications

(40 citation statements)

References 34 publications

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“…If Γ a (X, µ) is a measure-preserving action, then the pushforward measure (stab a ) * µ is an invariant random subgroup of Γ called the type of a. We extend ideas of Tucker-Drob from [13] to show the following.…”

Section: Introductionmentioning

confidence: 96%

Topology and convexity in the space of actions modulo weak equivalence

Burton¹

2017

Ergod. Th. Dynam. Sys.

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We analyse the structure of the quotient A∼(Γ, X, µ) of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We introduce a variant of an abstract construction of Fritz which encapsulates the convex combination operation on A∼(Γ, X, µ). This formalism allows us to define the geometric notion of an extreme point. We also discuss a topology on A∼(Γ, X, µ) due to Abert and Elek in which it is Polish and compact, and show that this topology is equivalent others defined in the literature. We show that the convex structure of A∼(Γ, X, µ) is compatible with the topology, and as a consequence deduce that A∼(Γ, X, µ) is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of A∼(Γ, X, µ) for amenable Γ by identifying it with the simplex of invariant random subgroups. In particular we conclude that A∼(Γ, X, µ) can be represented as a compact convex subset of a Banach space if and only if Γ is amenable. In the case of general Γ we prove a Krein-Milman type theorem asserting that finite convex combinations of the extreme points of A∼(Γ, X, µ) are dense in this space. We also consider the space A∼ s (Γ, X, µ) of stable weak equivalence classes and show that it can always be represented as a compact convex subset of a Banach space. In the case of a free group FN , we show that if one restricts to the compact convex set FR∼ s (FN , X, µ) ⊆ A∼ s (FN , X, µ) consisting of the stable weak equivalence classes of free actions, then the extreme points are dense in FR∼ s (FN , X, µ). * Research partially supported by NSF grant DMS-0968710 Theorem 1.2. A ∼ (Γ, X, µ) is equal to the closed convex hull of its extreme points. In other words, finite convex combinations of the extreme points of A ∼ (Γ, X, µ) are dense in A ∼ (Γ, X, µ).Given this result, it seems interesting to describe the extreme points of A ∼ (Γ, X, µ). In the amenable case, the identification with IRS(Γ) provides a complete such description, since the extreme points of IRS(Γ) are known to be the ergodic measures and consequently the extreme points of A ∼ (Γ, X, µ) for amenable Γ are exactly those actions with ergodic type. In the nonamenable case this description does not suffice. It is clear that any strongly ergodic action is an extreme point. We are able to show the following. Theorem 1.3. Suppose [a] ∈ A ∼ (Γ, X, µ) is an extreme point. Let a = Z a z dη(z) be the ergodic decomposition of a. Then there is a measure-preserving action b of Γ such that for η-almost all z ∈ Z we have [a z ] = [b].

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Section: Introductionmentioning

confidence: 96%

Topology and convexity in the space of actions modulo weak equivalence

Burton¹

2017

Ergod. Th. Dynam. Sys.

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“…This notion was introduced in [20]. In [37], it is proven that the definition given in this paper is equivalent to the one introduced in [20]. Some basic facts: all Bernoulli shifts over Γ are weakly equivalent.…”

Section: Weak Containmentmentioning

confidence: 90%

Zero Entropy Is Generic

Bowen

2016

Entropy

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Dan Rudolph showed that for an amenable group, Γ, the generic measure-preserving action of Γ on a Lebesgue space has zero entropy. Here, this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward's recent generalization of Sinai's Factor Theorem, the Gaboriau-Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other.

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“…We will use the notion of a random Bernoulli shift over an invariant random subgroup; we refer the reader to [T,Section 5.3 For y ∈ Y write H y = stab b (y). By [DG,Proposition 8] we have λ G/Hy κ b 0 for ν-almost every y ∈ Y .…”

Section: Proof Of Lemma 22mentioning

confidence: 99%

Invariant random subgroups and action versus representation maximality

Burton

Kechris

2017

Proc. Amer. Math. Soc.

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Problem 1.1. If a, b ∈ A(G, X, µ) are free, ergodic, does κ a 0 κ b 0 imply a b?We provide a negative answer below. The proof is based on a result about invariant random subgroups of G = F ∞ , the free group on a countably infinite set of generators, which might be of independent interest.If I is a countable set and α is an action of a countable group G on I, we will write s α for the corresponding generalized shift action on 2 I with the usual product measure, given by (s α (gWe also let λ α be the representation on ℓ 2 (I) given by (λ α (g)which we will denote by λ G/H . We call a subgroup H ≤ G with [G : H] = ∞ action-maximal if s G/H is action-maximal and representation-maximal if λ G/H is representationmaximal. It was shown in [K1] that there are H which are action-maximal and also H which are representation-maximal, for any non-abelian free group G.An invariant random subgroup (IRS) of G is a probability Borel measure on Sub(G), the compact space of subgroups of G, which is invariant under the (continuous) action of G on Sub(G) by conjugation. Denote by M G ⊆ Sub(G) the set of all H ≤ G that are both action-maximal and representation-maximal. We show the following:Using this and the result of Dudko-Grigorchuk [DG, Proposition 8], we then prove the following: Theorem 1.2. Let G = F ∞ . Then there exists a free, ergodic a ∈ A(G, X, µ) such that a is not action-maximal but κ a 0 is representation-maximal.

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Weak equivalence and non-classifiability of measure preserving actions

Cited by 20 publications

References 34 publications

Topology and convexity in the space of actions modulo weak equivalence

Topology and convexity in the space of actions modulo weak equivalence

Zero Entropy Is Generic

Invariant random subgroups and action versus representation maximality

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