Krieger&#39;s finite generator theorem for actions of countable groups Ⅱ

Brandon Seward

doi:10.3934/jmd.2019012

Article Contents

2019, Volume 15: 1-39. Doi: 10.3934/jmd.2019012

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Krieger's finite generator theorem for actions of countable groups Ⅱ

Brandon Seward

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

Received: May 23, 2018

Revised: November 2018

Published: February 2019

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Abstract

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.

Keywords:

Bernoulli shift,
isomorphism,
entropy,
sofic entropy,
generating partition,
generator,
Gottschalk's surjunctivity conjecture,
Kaplansky's direct finiteness conjecture.

Mathematics Subject Classification: Primary: 37A35; Secondary: 37A15.

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References

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