Krieger’s finite generator theorem for actions of countable groups III

Alpeev, Andrei; Seward, Brandon

doi:10.1017/etds.2020.89

Cited by 9 publications

(27 citation statements)

References 27 publications

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“…As sofic entropy is always bounded above by Rokhlin entropy [4,2], we have the following immediate corollary.…”

Section: Introductionmentioning

confidence: 81%

“…When G is amenable and the action is free, the relative Rokhlin entropy coincides with relative Kolmogorov-Sinai entropy [31,2]. Additionally, similar to the Rudolph-Weiss theorem [29], it is known that h Rok G (X, µ | F ) is invariant under orbit equivalences for which the orbit-change cocycle is F -measurable [31].…”

Section: Action and Let α Be A Pre-partition If β Is A Partition And βmentioning

confidence: 93%

“…Here we develop only those continuity properties which are essential to studying h Rok G (L G , λ G ). A comprehensive study of the various continuity properties of Rokhlin entropy will be presented in Part III [2]. The results in Part III will in particular cover the case of actions which are not necessarily ergodic.…”

Section: And H Rokmentioning

confidence: 99%

“…When G is amenable and the action is free, the Rokhlin entropy coincides with classical Kolmogorov-Sinai entropy [33,2]. Rokhlin entropy is thus a natural analog of classical entropy.…”

Section: Introductionmentioning

confidence: 98%

“…We previously noted that one always has h Rok G (L G , λ G ) ≤ H(L, λ). When G is sofic, Rokhlin entropy is bounded below by sofic entropy [4,2] and thus h Rok G (L G , λ G ) = H(L, λ) whenever G is sofic. Since the definition of Rokhlin entropy does not require the acting group to be sofic, the statement…”

Section: Introductionmentioning

confidence: 99%

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Krieger’s finite generator theorem for actions of countable groups I

Seward

2018

Invent. math.

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We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért-Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ii) Bernoulli shifts have completely positive Rokhlin entropy; and (iii) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.h Rok G (X, µ) = inf H(α|I ) : α is a countable partition and σ-alg G (α)∨I = B(X) , where I is the σ-algebra of G-invariant sets. In this paper, we will only be interested in ergodic actions, in which case the Rokhlin entropy simplifies to h Rok G (X, µ) = inf H(α) : α is a countable generating partition .

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“…As sofic entropy is always bounded above by Rokhlin entropy [4,2], we have the following immediate corollary.…”

Section: Introductionmentioning

confidence: 81%

Section: Action and Let α Be A Pre-partition If β Is A Partition And βmentioning

confidence: 93%

Section: And H Rokmentioning

confidence: 99%

“…When G is amenable and the action is free, the Rokhlin entropy coincides with classical Kolmogorov-Sinai entropy [33,2]. Rokhlin entropy is thus a natural analog of classical entropy.…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Krieger’s finite generator theorem for actions of countable groups I

Seward

2018

Invent. math.

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Affinity of the Arov Entropy

Gurevich

2018

Funct Anal Its Appl

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Cost, ℓ2-Betti numbers and the sofic entropy of some algebraic actions

Gaboriau

Seward

2019

JAMA

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In 1987, Ornstein and Weiss discovered that the Bernoulli 2-shift over the rank two free group factors onto the seemingly larger Bernoulli 4-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein-Weiss map for countable groups. We relate the increase in entropy to the cost and to the first ℓ 2 -Betti number of the group. More generally, we study coboundary maps arising from simplicial actions and, under certain assumptions, relate ℓ 2 -Betti numbers to the failure of the Yuzvinsky addition formula. This work is built upon a study of entropy theory for algebraic actions. We prove that for actions on profinite groups via continuous group automorphisms, topological sofic entropy is equal to measure sofic entropy with respect to Haar measure whenever the homoclinic subgroup is dense. For algebraic actions of residually finite groups we find sufficient conditions for the sofic entropy to be equal to the supremum exponential growth rate of periodic points.

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Krieger’s finite generator theorem for actions of countable groups III

Cited by 9 publications

References 27 publications

Krieger’s finite generator theorem for actions of countable groups I

Krieger’s finite generator theorem for actions of countable groups I

Affinity of the Arov Entropy

Cost, ℓ2-Betti numbers and the sofic entropy of some algebraic actions

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