Graph products and measure equivalence: classification, rigidity, and quantitative aspects

We study graph products of groups from the viewpoint of measured grouptheory. We first establish a full measure equivalence classification of graphproducts of countably infinite groups over finite simple graphs with notransvection and no partial conjugation. This finds applications to theirclassification up to commensurability, and up to isomorphism, and to the studyof their automorphism groups. We also derive structural properties of vonNeumann algebras associated to probability measure-preserving actions of graphproducts. Variations of the measure equivalence classification statement aregiven with fewer assumptions on the defining graphs. We also provide aquantified version of our measure equivalence classification theorem, thatkeeps track of the integrability of associated cocycles. As an application, wesolve an inverse problem in quantitative orbit equivalence for a large familyof right-angled Artin groups. We then establish several rigidity theorems.First, in the spirit of work of Monod-Shalom, we achieve rigidity in orbitequivalence for probability measure-preserving actions of graph products, uponimposing extra ergodicity assumptions. Second, we establish a sufficientcondition on the defining graph and on the vertex groups ensuring that a graphproduct G is rigid in measure equivalence among torsion-free groups (in thesense that every torsion-free countable group H which is measure equivalent toG, is in fact isomorphic to G). Using variations over the Higman groups as thevertex groups, we construct the first example of a group which is rigid inmeasure equivalence, but not in quasi-isometry, among torsion-free groups.