Invariant percolation and measured theory of nonamenable groups
@article{Houdayer2011InvariantPA, title={Invariant percolation and measured theory of nonamenable groups}, author={Cyril Houdayer}, journal={arXiv: Group Theory}, year={2011} }
Using percolation techniques, Gaboriau and Lyons recently proved that every countable, discrete, nonamenable group $\Gamma$ contains measurably the free group $\mathbf F_2$ on two generators: there exists a probability measure-preserving, essentially free, ergodic action of $\mathbf F_2$ on $([0, 1]^\Gamma, \lambda^\Gamma)$ such that almost every $\Gamma$-orbit of the Bernoulli shift splits into $\mathbf F_2$-orbits. A combination of this result and works of Ioana and Epstein shows that every…
18 Citations
von Neumann’s problem and extensions of non-amenable equivalence relations
- MathematicsGroups, Geometry, and Dynamics
- 2018
The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [GL07] to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable…
Finitary random interlacements and the Gaboriau–Lyons problem
- MathematicsGeometric and Functional Analysis
- 2019
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…
Solid Ergodicity and Orbit Equivalence Rigidity for Coinduced Actions
- Mathematics
- 2020
We prove that the solid ergodicity property is stable with respect to taking coinduction for a fairly large class of coinduced action. More precisely, assume that $\Sigma<\Gamma$ are countable groups…
Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions
- Mathematics
- 2011
Factors of IID on Trees
- MathematicsCombinatorics, Probability and Computing
- 2016
This work presents some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics and probability, and includes bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.
THE COMPLEXITY OF CLASSIFICATION PROBLEMS IN ERGODIC THEORY
- Mathematics
- 2011
theorems in various areas of mathematics. In the last three lectures, we will show how these ideas can be applied in proving a strong non-classication theorem for orbit equivalence. Given a countable…
Invariant coupling of determinantal measures on sofic groups
- MathematicsErgodic Theory and Dynamical Systems
- 2014
To any positive contraction $Q$ on $\ell ^{2}(W)$, there is associated a determinantal probability measure $\mathbf{P}^{Q}$ on $2^{W}$, where $W$ is a denumerable set. Let ${\rm\Gamma}$ be a…
Fixed points for bounded orbits in Hilbert spaces
- Mathematics
- 2015
Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes…
Solidity of Type III Bernoulli Crossed Products
- Mathematics
- 2016
AbstractWe generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra A0, any faithful normal state $${\varphi_0}$$φ0 and any discrete group…
Expanders have a spanning Lipschitz subgraph with large girth
- Mathematics
- 2013
We show that every regular graph with good local expansion has a spanning Lipschitz subgraph with large girth and minimum degree. In particular, this gives a finite analogue of the dynamical solution…
References
SHOWING 1-10 OF 92 REFERENCES
Group-invariant Percolation on Graphs
- Mathematics
- 1999
Abstract. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes…
SOME COMPUTATIONS OF 1-COHOMOLOGY GROUPS AND CONSTRUCTION OF NON-ORBIT-EQUIVALENT ACTIONS
- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2006
For each group $G$ having an infinite normal subgroup with the relative property (T) (e.g. $G=H\times K$, with $H$ infinite with property (T) and $K$ arbitrary) and each countable abelian group…
ORBIT INEQUIVALENT ACTIONS OF NON-AMENABLE GROUPS
- Mathematics
- 2008
Considertwo freemeasurepreservinggroupactions y (X,µ),� y (X,µ), and a measure preserving actiony a (Z,�) where (X,µ),(Z,�) are standard probability spaces. We show how to construct free measure…
An uncountable family of nonorbit equivalent actions of _
- Mathematics
- 2005
Recall that two ergodic probability measure preserving (p.m.p.) actions σi for i = 1, 2 of two countable groups Γi on probability measure standard Borel spaces (Xi, μi) are orbit equivalent (OE) if…
Ergodic theory of amenable group actions. I: The Rohlin lemma
- Mathematics
- 1980
Classically, ergodic theory began with the study of flows or actions of R. Later, for technical reasons, much of the theory was first developed for actions of Z. More recently, there has been…
Invariant percolation and harmonic Dirichlet functions
- Mathematics
- 2004
Abstract.The main goal of this paper is to answer Question 1.10 and settle Conjecture 1.11 of Benjamini–Lyons–Schramm [BenLS] relating harmonic Dirichlet functions on a graph to those on the infinite…
Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously
- Mathematics
- 1999
Abstract. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we…
On the superrigidity of malleable actions with spectral gap
- Mathematics
- 2006
Some of the most interesting aspects of the dynamics of measure preserving actions of countable groups on probability spaces, V rx (X, /?), are revealed by the study of group measure space von…
Non-orbit Equivalent Actions of F N
- Mathematics
- 2009
For any 2 ≤ n ≤ ∞, we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group F n. These actions arise as diagonal products between a generalized Bernoulli action…