Ergodic Subequivalence Relations Induced by a Bernoulli Action

Abstract

Let Γ be a countable group and denote by S the equivalence relation induced by the Bernoulli action Γ [0, 1] Γ , where [0, 1] Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation R of S, there exists a partition {X i } i≥0 of [0, 1] Γ into R-invariant measurable sets such that R |X0 is hyperfinite and R |Xi is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.

W*–superrigidity for Bernoulli actions of property (T) groups

“…The proof of [8], Lemma 5 provides a family {A i } i∈I of von Neumann subalgebras of M 1 and a family {Γ i } i∈I of finite subgroups of Γ 1 such that…”

The expected degree of minimal spanning forests

“…Now, the desired result is an immediate consequence of (2) and [3,Theorem 1.3] from the work of Epstein-Monod.…”

Relative property (T) for the subequivalence relations induced by the action of SL2(Z) on Ioana1 2010Advances in MathematicsSelf Cite16000View full textAdd to dashboardCite

“…It is not known whether rigidity implies strong ergodicity for equivalence relations (this question has been communicated to me by S. Popa). An affirmative answer to this question together with Theorem 3.1 would provide a different proof of N. Ozawa's result saying that any ergodic, nonhyperfinite subequivalence relation R of S is strongly ergodic (see [37] and [6]). …”

“…Thus, it is proven in [43, 5.2], that, if S is the equivalence relation induced by a Bernoulli action of a countable group, then the II 1 factor L(R) is prime, for any ergodic, non-hyperfinite subequivalence relation R of S. Moreover, as shown in [6], any such R is strongly ergodic (see [36] in the case of exact groups Γ ). Recently, N. Ozawa has shown that, in the context of 0.1, any ergodic subequivalence relation R is either hyperfinite or strongly ergodic [37].…”