Group Extensions for Random Shifts of Finite Type

The paper proves the equivalence h(r)Gur(T) = h(r)Gur(Tab) if and only if G is amenable for topologically mixing random shifts of finite type (Theorem 1.3), combining the general monotonicity h(r)Gur(T) ≤ h(r)Gur(Tab) from the definitions with a strict inequality when G is non-amenable (Theorem 7.1) and, for amenable G, a reverse inequality yielding equality (Theorem 1.2) . The candidate solution correctly observes the counting-based inequality chain and that Gab is amenable, but it crucially relies on an additional assertion that, in the random setting, amenability of the extension group implies equality of the extension’s relative Gurevič pressure with the base (and similarly for the abelianized extension). The uploaded paper does not establish this pressure equality direction; instead, it proves an equality/characterization at the level of the spectral radius of the random Perron–Frobenius operator (Theorem 1.9) and only one implication Π(r)Gur(f,ϕ)=Π(r)Gur(T,ϕ̃) ⇒ G amenable (Theorem 1.8, part II) . Thus, the model’s proof has a gap (it asserts amenability ⇒ Π(r)Gur(T,ϕ̃)=Π(r)Gur(f,ϕ) without justification in the random framework), even though its final conclusion matches the paper’s result.