Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry

Let G be a right-angled Artin group with jOut.G/j < C1.We prove that if a countable group H with bounded finite subgroups is measure equivalent to G, with an L 1 -integrable measure equivalence cocycle towards G, then H is finitely generated and quasi-isometric to G.In particular, through work of Kleiner and the second author, H acts properly and cocompactly on a CAT.0/ cube complex which is quasi-isometric to G and equivariantly projects to the right-angled building of G.As a consequence of work of the second author, we derive a superrigidity theorem in integrable measure equivalence for an infinite class of right-angled Artin groups, including those whose defining graph is an n-gon with n 5.In contrast, we also prove that if a right-angled Artin group G with jOut.G/j < C1 splits nontrivially as a product, then there does not exist any locally compact group which contains all groups H that are L 1 -measure equivalent to G as lattices, even up to replacing H by a finite-index subgroup and taking the quotient by a finite normal subgroup.