Weak containment in the space of actions of a free group
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.
scientific article; zbMATH DE number 6105481
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