Algebraic Properties of the Group of Germs of Diffeomorphisms

The paper proves [G^n_0,G^n_k]=G^n_k for all k≥1 via Poincaré linearization (choose A=λ·id with 0<|λ|<1; linearize hA by a ϕ∈G^n_k; then [ϕ,A]=h), and deduces [G^n_0,G^n_0]={f: det Df(0)=1} by writing f as a commutator with a linear map, see Theorem 2.5, Theorem 2.8, and Corollary 2.7 in the paper’s Section 2 , with definitions in Section 1 and the linearizing-map refinement in Remark 2.4 . The candidate solution proves the same statements but via a different route: a contraction-mapping solution of the cohomological equation g∘h=f∘h∘g for a linear contraction g, and a standard matrix-commutator argument for SL_n(C). The approach is sound; minor analytic details (completeness of the weighted Banach space, explicit invariance of the space under Φ, choosing r ensuring sup||Df||≤1+ε) are left at sketch level but are standard to fill in. Hence both are correct, with different proofs.