Free products, Orbit Equivalence and Measure Equivalence Rigidity

TL;DR

This paper explores the structure of orbit equivalence in free product decompositions of groups and equivalence relations, establishing rigidity results and invariants that lead to classification insights in orbit equivalence and II$_1$ factors.

Contribution

It introduces the notion of freely indecomposable (FI) equivalence relations, providing criteria based on non-hyperfiniteness and vanishing first $L^2$-Betti number, and proves rigidity results under weak assumptions.

Findings

Non-hyperfinite, FI relations are uniquely decomposable in free products.

Measure equivalences of free products are induced by component-wise measure equivalences.

New classification results in orbit equivalence and II$_1$ factors.

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.