A Markovian and Roe-algebraic Approach to Asymptotic Expansion in Measure

The paper proves the full equivalence among (1) asymptotic expansion, (2) quasi-locality of the Drutu–Nowak projection G, and (3) G being a norm limit of finite-propagation operators (Theorem E = Theorems 4.8 and 4.16) with precise bridges via dynamical quasi-locality/propagation and the Φ-lift to the warped cone, including the crucial Lemma 4.4 and Propositions 4.6, 4.7, and 4.15 . By contrast, the model’s proof outline contains two substantive gaps: (i) it asserts the existence of a single finitely supported symmetric measure μ on Γ whose averaging operator has a uniform spectral gap on L2_0(X,ν) for every asymptotically expanding action, which is not established here (the paper proceeds via Markov S-expansion and exhaustion rather than a single μ) ; and (ii) in the contrapositive for quasi-locality ⇒ expansion, it improperly transfers a small-expansion bound from A_k to an open superset U_k (monotonicity goes the wrong way), whereas the paper derives the implication using Proposition 4.7 together with Lemma 4.2 and compact approximation on fibers . Hence the paper’s argument is sound and complete for the stated setting, while the model’s derivation is flawed.