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.

minor additions

References in corpus (3)

• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• Relative Property (T) Actions and Trivial Outer Automorphism Groups
• Un theoreme de Kurosh pour les relations d'equivalence boreliennes

Cited by in corpus (6)

• A survey of Measured Group Theory
• Bass-Serre rigidity results in von Neumann algebras
• Measure Equivalence Rigidity and Bi-exactness of Groups
• Some $OE$ and $W^{\ast }$-rigidity results for actions by wreath product groups
• Measure equivalence classification of transvection-free right-angled Artin groups
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type