A measurable-group-theoretic solution to von Neumann’s problem

By Damien Gaboriau, Russell Lyons. 2007

141 citations recorded by the source corpus.

We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1a24c77a64dc0ab1b5831ddc3b3382e594ede4a

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Orbit Equivalence and Measured Group Theory (2010, background): “2 ([GL09]) For any non-amenable countable group Γ, the orbit equivalence relation of the Bernoulli shift action Γ y ([0, 1],Leb)Γ contains a subrelation generated by a free ergodic p.”
ORBIT INEQUIVALENT ACTIONS OF NON-AMENABLE GROUPS (2007, background): “Gaboriau and Lyons [ GL07 ] showed that every countable, non-amenable group admits a free, measure preserving, ergodic action on a standard probability space (X;„) so that the orbit equivalence relation induced by the action contains the orbit equivalence relation induced by a free, measure preservi”
Probability on Trees and Networks (2017, methodology): “In the context of the ergodic theory of equivalence relations mentioned above, it turns out that indistinguishability is equivalent to ergodicity of the equivalence relation given by the infinite clusters; see Gaboriau and Lyons (2009). Indistinguishability has a number of consequences, as shown by ”
Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$ (2006, background): “Subsequently, a combination of results, ideas from [9, 16] and the present paper has been used to show that the above result holds true for any non-amenable group [9].”
Unimodular random trees (2012, background): “ree action of a free group F 2 on 2 generators. If, instead, the cost is assumed only to be larger than 1, then there is a subrelation that is generated by a free action of F 2: see Proposition 14 of Gaboriau and Lyons (2009). In the remaining case where the treeable equivalence relation has cost 1,”