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Let M be a closed orientable C∞ riemannian manifold of dimension n+1 ≥ 3 with constant sectional curvature −1. It is covered by the hyperbolic space H, and a deck transformation is naturally extended to a C∞ diffeomorphism of the sphere at infinity, which is conformal with respect to the standard metric of the sphere. In consequence, we are provided with a canonical C∞ action of the fundamental group Γ = π1(M) on the n-sphere S, or, alternatively speaking, a homomorphism ρ0 of Γ into the diffeomorphism group Diff(S). Another action or homomorphism ρ : Γ → Diff(S) is said to be C-close to ρ0, if ρ(γ) is close to ρ0(γ) relative to the C-topology for finitely many γ ∈ Γ that altogether generate Γ (recall that Γ is finitely generated). One of the main results in the present note is

A remark on local rigidity of conformal actions on the sphere