Ratio limits and pressure function for group extensions of Gibbs Markov maps

The paper’s Proposition 6.1 is stated and proved correctly: for topologically transitive group extensions by amenable finitely generated groups, it shows P_φ(T_ψ) = P_φ(T_{ab,ψ}) and produces a subsequence n_i with P_φ(T_ψ) = P_φ(T) + lim_i (1/n_i) log λ(G, m_{n_i}). The candidate reproduces many key constructions (P_n on ℓ^2(G), normalization via m_n, and the convolution identity), but (i) overclaims an exact limit identity for fixed n and a “sup over n” pressure formula not proved in the paper, (ii) appeals to a Dougall–Sharp SFT result to deduce equality of pressures for general GM systems via ‘coding’ without justifying the extra hypotheses, and (iii) omits crucial assumptions used in the paper (notably that G is finitely generated and ψ is constant on partition elements in the FG case). Hence the model solution contains material errors/overreach, whereas the paper’s proof is careful and correct.