Free products, Orbit Equivalence and Measure Equivalence Rigidity - ENS de Lyon - École normale supérieure de Lyon Access content directly
Journal Articles Groups, Geometry, and Dynamics Year : 2012

Free products, Orbit Equivalence and Measure Equivalence Rigidity


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.
Fichier principal
Vignette du fichier
Alva-Gabo-2-prunel.pdf (338.2 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

ensl-00288583 , version 1 (17-06-2008)
ensl-00288583 , version 2 (17-02-2009)



Aurélien Alvarez, Damien Gaboriau. Free products, Orbit Equivalence and Measure Equivalence Rigidity. Groups, Geometry, and Dynamics, 2012, 6 (1), pp.53-82. ⟨10.4171/GGD/150⟩. ⟨ensl-00288583v2⟩
216 View
215 Download



Gmail Facebook Twitter LinkedIn More