Non-geometric property (T) of warped cones

The paper proves that for a finitely generated dense subgroup G of a compact Lie group M (acting by left translations), the associated warped system fails geometric property (T) for every sequence t(n)→∞. The authors fix the admissible measure µ_t(n)=t(n)^mµ on each component, construct a coarse generating gordo set E_r, derive the identity Δ_{E_r}=|S|ϕ−(|S|−Δ_G)(ϕ−L_r), and then show L_r has no spectral gap in an appropriate quotient via heat-kernel comparison; this is transferred to Δ_{E_r} in the maximal Roe algebra using a crossed-product representation and density of G in M, completing the proof (Theorem 1.2/4.1, Lemma 2.12, Lemma 3.1, Proposition 3.2 with Lemma 3.3, Lemma 4.2, and the diagram around Ψ) . By contrast, the candidate solution normalizes Haar measure separately on each component and concludes there is no gordo set and that spectral gap fails in the canonical L2-representation because operator norms decay like t(n)^{-m}. This contradicts the admissible-measure framework used in Winkel’s definition (which requires existence of a gordo set and is measure independent among admissible choices) and fails under the paper’s µ_t(n)=t(n)^mµ, where small-ball volumes are uniformly bounded below and E_r is gordo (Definition 2.7) . The candidate’s Schur estimate and kernel-equality arguments therefore do not apply to the setting analyzed in the paper’s proof.