Weak containment in the space of actions of a free group
type: Item
instance of: scholarly article
title: Weak containment in the space of actions of a free group
zbMATH Open document ID: 1273.37003
author: Alexander S. Kechris
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
full work available at URL: https://authors.library.caltech.edu/32467/
review text: Let \(\Gamma\) be a countable group acting on a non-atomic standard Borel probability space \((X,\mu)\). \(A(\Gamma, X, \mu)\) denotes the set of all such actions and \(\gamma^a(x)=a(\gamma, x)\) denotes the corresponding automorphism of \((X, \mu)\). Given \(a\in A(\Gamma, X, \mu)\) and \(b\in A(\Gamma, Y, \nu)\), we say that \(a\) is weakly contained in \(b\) if for any Borel sets \(A_1, \ldots , A_n\) in \(X\), \(\gamma_1, \ldots, \gamma_m \in \Gamma\), \(\epsilon > 0\), there exists Borel sets \(B_1, \ldots , B_n\) in \(X\) such that \[ |\mu(\gamma^a_i(A_j)\cap A_k)-\nu(\gamma_i^b(B_j)\cap B_k)|<\epsilon ,\quad\forall i\leq m,\quad\forall j, k\leq n. \] This defines a partial order giving a maximal element \(a^{\infty}\) which has a dense conjugacy class in \(A(\Gamma, X, \mu)\); see [\textit{E. Glasner} et al., Bull. Lond. Math. Soc. 38, No. 6, 932--936 (2006; Zbl 1116.28013)]. For \(\Gamma= {\mathbb F}_n\), the free group on \(n\) generators, it is shown that the action \(p_{{\mathbb F}_n}\) is maximal in the partial order (where \(p_{\Gamma}\) is the left translation action of \(\Gamma\) on \(\hat{\Gamma}\), a measure preserving action of \(\Gamma\)). Applications to the theory of costs are given (see the articles by \textit{D. Gaboriau} [Notices Am. Math. Soc. 57, No. 10, 1295--1296 (2010; Zbl 1226.37001)] and \textit{M. Abért} and \textit{N. Nikolov} [J. Eur. Math. Soc. (JEMS) 14, No. 5, 1657--1677 (2012; Zbl 1271.57046)]), to give a new proof of a result due to \textit{D. Gaboriau} [Invent. Math. 139, No. 1, 41--98 (2000; Zbl 0939.28012)], which predicts that the cost in \({\mathbb F}_n\) is equal to \(n\). A study of residually finite, countable groups for which the finite actions are dense in the space of measure preserving actions is given.
reviewed by: Geoffrey R. Goodson
Mathematics Subject Classification ID: 37A15
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37B05
zbMATH DE Number: 6105481
zbMATH Keywords: cost
zbMATH Keywords: free group action
zbMATH Keywords: weak containment
MaRDI profile type: Publication
OpenAlex ID: W1963528753
cites work: Rank gradient, cost of groups and the rank versus Heegaard genus problem
cites work: Bernoulli actions are weakly contained in any free action
cites work: Irreducibility of unitary group representations and reproducing kernels Hilbert spaces. Appendix on two point homogeneous compact ultrametric spaces in collaboration with Rostislav Grigorchuk.
cites work: Ergodic subequivalence relations induced by a Bernoulli action
cites work: Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. I
cites work: Cost of equivalence relations and groups
cites work: Amenable actions, free products and a fixed point property
cites work: EVERY COUNTABLE GROUP HAS THE WEAK ROHLIN PROPERTY
cites work: Q5718639
cites work: A converse to Dye’s theorem
cites work: Mixing actions of groups with the Haagerup approximation property
cites work: Relative property (T) for the subequivalence relations induced by the action of SL\(_2(\mathbb Z)\) on \(\mathbb T^2\)
cites work: Amalgamated free products of weakly rigid factors and calculation of their symmetry groups
cites work: Q3407797
cites work: Topics in orbit equivalence
cites work: Amenable actions and almost invariant sets
cites work: Q4656878
cites work: An example of a solid von Neumann algebra
cites work: Measures invariant under actions of \(F_ 2\)
cites work: Q5810565
cites work: Q3687381
DOI: 10.1007/S11856-011-0182-6
description: scientific article; zbMATH DE number 6105481