The space of stable weak equivalence classes of measure-preserving actions | Academic Article individual record

© 2017 Elsevier Inc. The concept of (stable) weak containment for measure-preserving actions of a countable group Γ is analogous to the classical notion of (stable) weak containment of unitary representations. If Γ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Γ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when Γ=Z this simplex has only a countable set of extreme points but when Γ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Γ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points.

author list (cited authors)
Bowen, L., & Tucker-Drob, R.
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Elsevier bv Publisher
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