Bernoulli actions of type III_1 and L^2-cohomology

Stefaan Vaes; Jonas Wahl

Bernoulli actions of type III_1 and L^2-cohomology

Dynamical Systems 2018-04-24 v4 Group Theory Operator Algebras

Authors: Stefaan Vaes , Jonas Wahl

Abstract

We conjecture that a countable group $G$ admits a nonsingular Bernoulli action of type III $_1$ if and only if the first $L^2$ -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 III $_1$ Bernoulli actions of the group of integers and the free groups, with different degrees of ergodicity.

Keywords

Bernoulli actions type III_1 L^2-cohomology countable groups nonsingular actions ergodicity

Cite

@article{arxiv.1705.00439,
  title  = {Bernoulli actions of type III_1 and L^2-cohomology},
  author = {Stefaan Vaes and Jonas Wahl},
  journal= {arXiv preprint arXiv:1705.00439},
  year   = {2018}
}

Comments

v4: minor changes, final version, to appear in Geometric and Functional Analysis (GAFA)

Bernoulli actions of type III_1 and L^2-cohomology

Abstract

Keywords

Cite

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