Free products Orbit Equivalence and Measure Equivalence Rigidity

Niveau: Supérieur, Licence, Bac+2
Free products, Orbit Equivalence and Measure Equivalence Rigidity Aurélien Alvarez and Damien Gaboriau? February 17, 2009 Abstract We study the analogue in orbit equivalence of free product decomposi- tion and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable (FI) standard prob- ability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first L2-Betti num- ber. We obtain Bass-Serre rigidity results, 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 ergodic- ity hypothesis for the components. We deduce, for instance, that a measure equivalence between two free products of non-amenable groups with vanishing first 2-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in Orbit Equivalence and II1 factors. 1 Introduction Bass-Serre theory [Ser77] studies groups acting on trees and offers extremely power- ful tools to understand their structure, together with a geometric point of view that illuminates several classical results on free product decompositions. For instance Kurosh's subgroup theorem [Kur34], that describes the subgroups in a free product of groups and, as a by-product, the essential uniqueness in free product decomposi- tions into freely indecomposable subgroups, is much easier to handle via Bass-Serre theory.

• free products


• group


• ?2 ?


• standard borel


• theorem there


• equivalence relations


• measure preserving


• trivial finite direct


• actions ?