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Bernoulli actions of type III1 and L2-cohomology

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Abstract

We conjecture that a countable group G admits a nonsingular Bernoulli action of type III1 if and only if the first L2-cohomology of G is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III1 Bernoulli actions of the groups \({\mathbb{Z}}\) and the free groups \({\mathbb{F}_n}\), with different degrees of ergodicity.

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Authors and Affiliations

  1. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, box 2400, 3001, Leuven, Belgium

    Stefaan Vaes & Jonas Wahl

Authors
  1. Stefaan Vaes
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  2. Jonas Wahl
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Correspondence to Stefaan Vaes.

Additional information

SV and JW are supported by European Research Council Consolidator Grant 614195 RIGIDITY, and by long term structural funding – Methusalem grant of the Flemish Government.

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Vaes, S., Wahl, J. Bernoulli actions of type III1 and L2-cohomology. Geom. Funct. Anal. 28, 518–562 (2018). https://doi.org/10.1007/s00039-018-0438-y

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  • DOI: https://doi.org/10.1007/s00039-018-0438-y

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