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Free products, Orbit Equivalence and Mea...
Alvarez, Aurélien...

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Free products, Orbit Equivalence and Measure Equivalence Rigidity by Alvarez, Aurélien ( Author )
Lemonjar Open Access
N.A
17-06-2008
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 L2-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 ℓ2-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in Orbit Equivalence and II1 factors.
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English
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doi:10.4171/GGD/150
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