Ergodicity and type of nonsingular Bernoulli actions

Michael Björklund, Zemer Kosloff, Stefaan Vaes

Research output: Contribution to journalArticlepeer-review

Abstract

We determine the Krieger type of nonsingular Bernoulli actions G↷ ∏ gG({ 0 , 1 } , μg). When G is abelian, we do this for arbitrary marginal measures μg. We prove in particular that the action is never of type II if G is abelian and not locally finite, answering Krengel’s question for G= Z. When G is locally finite, we prove that type II does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II1 or III1. When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of Vaes and Wahl (Geom Funct Anal 28:518–562, 2018) by proving that a group G admits a Bernoulli action of type III1 if and only if G has nontrivial first L2-cohomology.

Original languageAmerican English
Pages (from-to)573-625
Number of pages53
JournalInventiones Mathematicae
Volume224
Issue number2
DOIs
StatePublished - May 2021

All Science Journal Classification (ASJC) codes

  • General Mathematics

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