Abstract
Abstract We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.
Highlights
Let us say that a free ergodic p.m.p. action of G is Schmidt if the associated orbit equivalence relation admits a non-trivial central sequence in its full group
A countable group, being a discrete p.m.p. groupoid on a singleton, is never Schmidt.) The following lemma implies that the Schmidt property of G follows once we find a free p.m.p
We show that the sequence (Tn)◦f ∈ C if f belongs to the set C1, which is slightly smaller than C, of all elements (x,mi=1) ∈ X ×
Summary
Independent of whether the FC-center of G has finite or infinite center, the above construction yields a p.m.p. action G (W , ω) and a central sequence (Tn) in the full group of the translation groupoid G ⋉ (W , ω). This construction is flexible enough to apply to the more general set-up, and we are able to deduce the Schmidt property for all groups with infinite FC-center
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