von Neumann’s problem and extensions of non-amenable equivalence relations

Abstract

The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [GL07] to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pmp) equivalence relation R, the Bernoulli extension over a non-atomic base space (K, κ) contains the orbit equivalence relation of a free ergodic pmp action of F2. Moreover, we provide conditions which imply that this holds for any non-trivial probability space K. Second, we use this result to prove that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).F 2 by [GL07, Theorem 1], it follows that the same is true for R K . However, Theorem A is new whenever R does not arise as the orbit equivalence relation of a free pmp action of a countable group (see [Fu99] for examples of such R). Also, note that if R = R(Γ [0, 1] Γ ), then R K is isomorphic to R. Theorem A implies that R contains the orbits of a free ergodic pmp action of F 2 , for any non-amenable Γ, and therefore recovers [GL07, Theorem 1].Remark 1.2. At the end of [GL07], the authors posed the following analogue of von Neumann's problem for equivalence relations: does every ergodic non-amenable countable pmp equivalence relation R contain R(F 2 X) for some free ergodic pmp action of F 2 ? The main result of [GL07] shows that this is indeed the case if R arises from the Bernoulli action with base ([0, 1], λ) of a non-amenable countable group. Theorem A shows that, more generally, this holds for the Bernoulli extension with base ([0, 1], λ) of any ergodic non-amenable countable pmp equivalence relation.We turn now to the second main result of this paper and to the history motivating it. In the early 1980s, D. Ornstein and B. Weiss [OW80], extending work of H. Dye [Dy59], showed that any two ergodic pmp actions of countable infinite amenable groups are orbit equivalent. Moreover, as a consequence of [CFW81], all free properly ergodic pmp actions of a unimodular amenable lcsc group G are pairwise orbit equivalent. On the other hand, over the next two decades, several families of non-amenable countable groups, including property (T) groups [Hj02] and non-abelian free groups [GP03], were shown to admit uncountably many actions which are pairwise not orbit equivalent.Unifying many of these results, it was shown in [Io06] that any countable group Γ containing a copy of F 2 has uncountably many free ergodic actions which are pairwise not orbit equivalent. Thus nearly three decades after the solution to von Neumann's problem [Ol80], the relationship between general non-amenable groups and the prototypical example of F 2 came again into focus. Gaboriau and Lyons' result in [GL07] was followed shortly by [Ep07], in which I. Epstein combined [GL07] with the methods of [Io06] via a new co-induction construction for group actions, proving that any countable non-amenable group Γ admits unc...

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