Geodesic
von Neumann’s problem and extensions of non-amenable equivalence relations
Lewis Bowen1 ; Daniel Hoff2 ; Adrian Ioana3 orcid
1University of Texas at Austin, USA
2University of California Los Angeles, USA
3University of California San Diego, La Jolla, USA
Groups, geometry, and dynamics, Tome 12 (2018) no. 2, pp. 399-448

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DOI

The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [17] 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).
DOI : 10.4171/ggd/456
Classification : 37-XX
Mots-clés : Ergodic equivalence relations, percolation, Bernoulli extension, orbit equivalence, von Neumann equivalence

Lewis Bowen  1   ; Daniel Hoff  2   ; Adrian Ioana  3

1 University of Texas at Austin, USA
2 University of California Los Angeles, USA
3 University of California San Diego, La Jolla, USA 
Lewis Bowen; Daniel Hoff; Adrian Ioana. von Neumann’s problem and extensions of non-amenable equivalence relations. Groups, geometry, and dynamics, Tome 12 (2018) no. 2, pp. 399-448. doi: 10.4171/ggd/456
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