2017/05/09 by Lewis Bowen, Bowen, Lewis, Robin Tucker-Drob +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1705.03528
openalex publication_date 2017/05/09 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
The concept of (stable) weak containment for measure-preserving actions of a\ncountable group \Γ is analogous to the classical notion of (stable) weak\ncontainment of unitary representations. If \Γ is amenable then the\nRokhlin lemma shows that all essentially free actions are weakly equivalent.\nHowever if \Γ is non-amenable then there can be many different weak and\nstable weak equivalence classes. Our main result is that the set of stable weak\nequivalence classes naturally admits the structure of a Choquet simplex. For\nexample, when \Γ=\ℤ this simplex has only a countable set of\nextreme points but when \Γ is a nonamenable free group, this simplex is\nthe Poulsen simplex. We also show that when \Γ contains a nonabelian free\ngroup, this simplex has uncountably many strongly ergodic essentially free\nextreme points.\n