Free products, Orbit Equivalence and Measure Equivalence Rigidity

Abstract : We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ({\FI}) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first $L^2$-Betti number. We obtain Bass-Serre rigidity results, \textit{i.e.} forms of uniqueness in free product decompositions of equivalence relations with ({\FI}) components. The main features of our work are weak algebraic assumptions and no ergodicity hypothesis for the components. We deduce, for instance, that a measure equivalence between two free products of non-amenable groups with vanishing first $\ell^2$-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in Orbit Equivalence and II$_1$ factors.
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Aurélien Alvarez, Damien Gaboriau. Free products, Orbit Equivalence and Measure Equivalence Rigidity. Groups Geometry and Dynamics, European Mathematical Society, 2012, 6 (1), pp.53-82. ⟨10.4171/GGD/150⟩. ⟨ensl-00288583v2⟩

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