On closed manifolds admitting an Anosov diffeomorphism but no expanding map

The paper proves that every infra-nilmanifold of dimension < 12 with an Anosov diffeomorphism also admits an expanding map (Theorem 6.1), by showing that every Anosov rational nilpotent Lie algebra of dimension < 12 admits a positive grading, via a rank/Galois analysis and the key “shift-by-d_A” bracket lemma (Proposition 5.4) . The candidate solution generally tracks this strategy, but it makes a substantive error in the n1 = 6 case: it claims there are no Anosov Lie algebras of type (6,2,·), whereas the paper shows type (6,2,2) does occur and is positively graded (Proposition 6.10) . It also slightly misstates the bracket-confinement parameter as depending on “rank” rather than on d_A from ker φ_A (Proposition 5.4) , and needlessly treats n1 = 2 though Anosov Lie algebras satisfy n1 ≥ 3 in this range . The paper’s case analyses for n1 = 3 (Proposition 6.2), n1 = 4 (Proposition 6.6), n1 = 5 (Corollary 6.8), n1 = 6 (Propositions 6.10–6.11), and n1 = 7 (Proposition 6.14) are consistent and complete, while the model’s account misclassifies the n1 = 6, type (6,2,2) case .