Partially Hyperbolic Lattice Actions on 2-Step Nilmanifolds

The paper proves topological and smooth rigidity for higher-rank lattice actions under a QI–partially hyperbolic hypothesis without assuming a global fiber trivialization; it builds an equivariant fiber map to a hyperbolic toral automorphism and uses Avila–Viana’s invariance principle and Property (T) cocycle trivialization to obtain a global conjugacy (Theorems 2.14, 2.15, 3.12, and the proof of Theorem 1.2). By contrast, the model’s argument crucially assumes the existence of a continuous global section of the S^1-bundle to write the action as a skew product with a continuous circle cocycle and then applies Livšic. Such a global section need not exist for the relevant nilmanifolds (e.g., the Heisenberg case), so the cohomological reduction as written is not justified. With this missing hypothesis fixed (trivial circle bundle), the remainder aligns in spirit with the paper’s conclusion, but as stated the model’s proof has a fatal gap.