ON THE TRIVIALIZABILITY OF RANK-ONE COCYCLES WITH AN INVARIANT FIELD OF PROJECTIVE MEASURES

Savini’s paper proves trivializability by: (i) pushing the W-invariant projective-measure field into a single H-orbit and hence a σ-equivariant map to an algebraic homogeneous space H/L (Theorem 5.2), (ii) using the initial-object machinery of T-algebraic representations and the compatibility hypothesis to force the algebraic hull J to be trivial in the real case (and to a specific central unipotent in the complex non-compatible case), (iii) gluing along P and NG(A) to obtain a continuous homomorphism Υ: G→H, and (iv) untwisting the cocycle to Υ|Γ. These steps are all present and justified in the paper, including the complex-hyperbolic detour via boundary maps when H(κ)=PU(n,1) (Proposition 6.4) . By contrast, the model’s outline hinges on an unproved—and in this setting unnecessary—claim that almost every fiber measure is Dirac via proximality. The paper deliberately avoids this, working instead with H/L and the algebraic-representations framework. The model also glosses over the precise use of compatibility and the gluing step needed to produce a homomorphism G→H. Hence, while the model’s high-level trajectory resembles the paper’s, its key steps are unsupported here.