One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II₁ factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II₁ factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.