Spectral Gap for Products and a Strong Normal Subgroup Theorem

The paper’s theorem and proof are internally consistent and supported by precise lemmas (notably Lemma 3.1 and Proposition 4.1) culminating in Theorem 1.6 / 4.9 on spectral gap for product actions. The candidate solution attempts a simpler direct argument via a uniform compact compensator and an L2 energy-transfer inequality, but both key steps are flawed: (i) the claimed openness of the set {H: H ∩ (K × {s}) ≠ ∅} in the Chabauty topology is incorrect, undermining the compactness argument used to get a uniform C; and (ii) the proposed inequality that integrates a pointwise choice c_s(x) and then upper-bounds by sup_{c∈C} norms is false in general (integral of a pointwise sup can exceed the sup of integrals). The paper’s approach avoids these pitfalls by introducing smoothing, G-uniform sequences, and a limit measure on Sub(G), from which asymptotic G2-invariance follows, contradicting the discordant-sequence lemma and yielding spectral gap.