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Geometric and Functional Analysis

, Volume 29, Issue 3, pp 659–689 | Cite as

Finitary random interlacements and the Gaboriau–Lyons problem

  • Lewis BowenEmail author
Article
  • 37 Downloads

Abstract

The von Neumann–Day problem asks whether every non-amenable group contains a non-abelian free group. It was answered in the negative by Ol’shanskii in the 1980s. The measurable version (formulated by Gaboriau–Lyons) asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau–Lyons. The proof uses an approximation to the random interlacement process by random multisets of geometrically-killed random walk paths. There are two applications: (1) the Gaboriau–Lyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all Bernoulli shifts factor onto each other.

Keywords and phrases

Random interlacement Von Neumann–Day problem Gaboriau–Lyons Non-amenable groups Bernoulli shifts 

Mathematics Subject Classification

37A35 

Notes

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Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Texas at AustinAustinUSA

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