Finitary random interlacements and the Gaboriau-Lyons problem
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 multistep 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.
Cite
@article{arxiv.1707.09573,
title = {Finitary random interlacements and the Gaboriau-Lyons problem},
author = {Lewis Bowen},
journal= {arXiv preprint arXiv:1707.09573},
year = {2019}
}
Comments
This version includes a proof that the FRI converges to the RI in distribution as T-> infinity. Also the proof of the hardest part of the main theorem has been simplified (following suggestions by an anonymous reviewer). And some unnecessary hypotheses in the main theorem have been removed