JournalsggdVol. 6, No. 1pp. 53–82

Free products, orbit equivalence and measure equivalence rigidity

  • Aurélien Alvarez

    Université d'Orléans, France
  • Damien Gaboriau

    École Normale Supérieure de Lyon, France
Free products, orbit equivalence and measure equivalence rigidity cover

Abstract

We study the analogue, in orbit equivalence, of free product decompositions and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable (FI\mathcal{FI}) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first L2L^2-Betti number. We obtain Bass–Serre rigidity results, i.e. forms of uniqueness in free product decompositions of equivalence relations with (FI\mathcal{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 2\ell^2-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in orbit equivalence and II1_1 factors.

Cite this article

Aurélien Alvarez, Damien Gaboriau, Free products, orbit equivalence and measure equivalence rigidity. Groups Geom. Dyn. 6 (2012), no. 1, pp. 53–82

DOI 10.4171/GGD/150