The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons  to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pmp) equivalence relation , the Bernoulli extension over a non-atomic base space contains the orbit equivalence relation of a free ergodic pmp action of . Moreover, we provide conditions which imply that this holds for any non-trivial probability space . 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).
Cite this article
Lewis Bowen, Daniel Hoff, Adrian Ioana, von Neumann’s problem and extensions of non-amenable equivalence relations. Groups Geom. Dyn. 12 (2018), no. 2, pp. 399–448DOI 10.4171/GGD/456