Generalized Bowen-Franks Groups and Profinite Conjugacy of Hyperbolic Toral Automorphisms

The paper proves that profinite conjugacy over Q (Q ∈ {Qp, Qa}) implies strong Bowen–Franks (BF) equivalence by identifying, for each g ∈ Q, the kernel of the canonical projection ϕg,*: GQ,* → Gg,* with the (closed) image of the endomorphism g(Γ*), and then showing any profinite conjugacy ΨQ carries these kernels to each other; modding out yields BFg(A) ≅R BFg(B) (Theorem 4.13, using Lemma 4.12) . The R-module structures and the pro-C limits GQ,* are set up in Subsection 4.2 and Definition 4.6 . By contrast, the model’s argument builds short exact sequences at finite levels and passes to inverse limits to get 0 → Kg,* → GQ,* → BFg(*) → 0, then asserts that an R-linear profinite conjugacy ΨQ maps Kg,A onto Kg,B and descends to the quotients. The key step—ΨQ(Kg,A) = Kg,B—is not justified: Kg,* was defined as lim← im(μg,h,*), and an arbitrary isomorphism of inverse limits need not preserve such inverse-limit-of-images submodules unless one identifies Kg,* with the (closed) image of g(Γ*) on GQ,*, as the paper does via Lemmas 4.8–4.11 . Without that identification (or a proof that lim← im(μg,h,*) equals im g(Γ*) in this setting), the model’s proof has a gap. Hence the paper’s proof is correct and complete, while the model’s solution is incomplete.