BOUNDARY ACTIONS BY HIGHER-RANK LATTICES: CLASSIFICATION AND EMBEDDING IN LOW DIMENSIONS, LOCAL RIGIDITY, SMOOTH FACTORS

The paper’s Theorem 4.6 precisely establishes a finite-to-one C^r, Γ–equivariant covering h: M → G/Q under assumptions (a)–(d), and its proof in §8.6.1 is complete: it invokes Theorem 8.1 to build a measurable G–equivariant map and lamination, uses a dimension count to show G–orbits are open, and concludes M is a single G–orbit so that h is a finite covering (see Theorem 4.6 statement and its proof outline, and Theorem 8.1 for the lamination and equivariant map construction ). In contrast, the candidate solution assumes uniform fiberwise contraction by a∈A, which is stronger than hypothesis (c) (negativity of Lyapunov exponents) and is not given by the theorem; the paper works in the nonuniform setting using nonstationary normal forms and PSR (subresonant) structures (see §§9.3–9.6 ). The candidate also appeals to a uniqueness-of-normal-forms identification with a homogeneous model without addressing the nonuniform coordinate issues that the paper handles via τ_q atlases and PSR maps (see §8.2 and Claim 8.4 ). While the candidate’s end conclusion matches the theorem, key steps hinge on unproven uniformity and an unsubstantiated semigroup-equivariance claim, so the model proof is not valid under the stated hypotheses.