Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations

The paper’s Theorem 2.4 proves that two actions α, β: G → N[T] are strongly cocycle conjugate iff Nα = Nβ and mod(α) = mod(β), with definitions and setup given in Definition 2.2 and the fundamental homomorphism facts Ker(mod) = cl_u([T]) and surjectivity (as recorded in the preliminaries) . The proof proceeds via ultraproduct Rohlin techniques (Theorem 3.1), cohomology-vanishing and quantitative patching lemmas (Section 4), and a diagonal intertwining culminating in Theorems 5.3 and 5.5 and the final proof of Theorem 2.4 . By contrast, the candidate solution makes two critical algebraic errors: (i) it misidentifies r(g)=αgβg^{-1} as a 1-cocycle for α̂ (it is for β̂), and (ii) it incorrectly rearranges r(g)=θ β̂_g(θ^{-1}) v(g) to conclude αg=θβgθ^{-1}v(g) and hence v(g)αg=θβgθ^{-1}. It also assumes, without proof, a strong “rectification lemma” that any cl_u([T])-valued 1-cocycle is cohomologous via θ∈[T] to a [T]-valued 1-cocycle. The paper achieves the needed rectifications by a different and rigorous route (ultraproduct Rohlin, quantitative cohomology vanishing), so the paper is correct while the model’s proof outline is flawed.