Free products, Orbit Equivalence and Measure Equivalence Rigidity

Aurélien Alvarez (1) , Damien Gaboriau (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Keywords: Kurosh's theorem; free indecomposability; Measure Equivalence; free products; Orbit Equivalence; Bass-Serre theory; $\ell^2$-Betti numbers

Domains: Mathematics [math]; Group Theory [math.GR]; Mathematics [math]; Operator Algebras [math.OA]

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⟩