A GROUP WITH PROPERTY (T) ACTING ON THE CIRCLE

The paper proves Theorem 1.2 by constructing an action of G = K(A3) on S1 via a Carathéodory loop ϕ: S1→D3, defining ρ(g) on the dense set ϕ^{-1}(Br(D3)), extending uniquely by an order-preserving lemma, and establishing the semiconjugacy ϕ(ρ(g)z) = gϕ(z) for all z ∈ S1 (definition and extension: , ). Proposition 3.6 shows ρ is a topological embedding; using completeness of the Polish group G yields that ρ(G) is closed (stated in the proof: ). Non-elementarity follows from minimality and strong proximality (Lemma 3.7 and Proposition 3.8: ). By contrast, the candidate solution replaces ϕ by an ad hoc coding map π built from a triadic Cantor model and defines ρ(g) to be affine on gaps. Two critical gaps result: (i) the claimed equivariance π∘ρ(g)=g∘π on each gap is not justified by ‘linear reparameterizations’ and generally fails unless ρ is defined using π; (ii) the closedness argument via a set-theoretic section σ assumes uniform limits f preserve π-fibers (so that π(f(σ(x))) is independent of σ), which is not established. Hence the model’s proof is incorrect, while the paper’s argument is sound.