Critical Exponent Rigidity for Θ−positive Representations

The paper proves Theorem 1.2—δαρ(Γ) ≤ 1 for all α ∈ Θ, with equality exactly for lattices and strict inequality for geometrically finite non‑lattices—via the regular distortion property, shadow estimates, and a doubling argument; see Theorem 4.2 (regular distortion), Proposition 4.5 and Corollary 4.6 (shadow bounds), Proposition 6.3 (Λ(Γ)=S1 gives δα ≤ 1), Theorem 6.4 (lower bound via dimension for lattices), and Proposition 7.1 + Proposition 7.2 (doubling and strict entropy drop) . The candidate solution hinges on an unsubstantiated inequality comparing a hypothesized α–cross ratio to the Möbius cross ratio, implying α(κ(ρ(γ))) ≥ ℓ(γ); no such cross‑ratio framework or inequality is established in the paper, and the authors instead need new tools (regular distortion and shadows). The pressure-based argument for the lattice case is also not justified for cusped lattices. Hence the model’s proof is not valid, while the paper’s argument is coherent and correct.