Charmenability and Stiffness of Arithmetic Groups

The paper’s Theorem A classifies when arithmetic subgroups of a connected Q‑algebraic group G are charmenable, in terms of the semisimple quotient S = G/R: either all arithmetic subgroups are charmenable or none are; the positive case occurs exactly when real rank(S) ≠ 1 and S has at most one R‑isotropic Q‑simple factor. The proof proceeds via a boundary criterion (Criterion 5.4), verifying condition (2) uniformly for all arithmetic groups using stiffness on the characters of the amenable radical (Proposition 5.8, built on Theorem C), and checking condition (1) in higher rank via the BH/BBHP singularity machinery (Proposition 5.7), with non‑charmenability in the excluded cases deduced from the normal‑subgroup dichotomy for charmenable groups. All these elements appear explicitly in the paper’s proof of Theorem A and supporting statements. The candidate solution follows the same blueprint and reaches the same classification. Minor issues: it identifies Rad(Λ) with Λ ∩ R(Q), whereas the paper shows Λ ∩ R(Q) is finite index in Rad(Λ), and it overstates that the boundary singularity condition (1) ‘holds exactly when’ the rank/product restriction holds (the paper uses Criterion 5.4 as a sufficient tool, not an equivalence). Otherwise, the alignment is strong. See Theorem A, Criterion 5.4, Proposition 5.7, Proposition 5.8, and the proof of Theorem A in the uploaded PDF ; the amenable radical structure is recorded in Proposition 4.9 and Corollary 4.10 , and the normal subgroup dichotomy is summarized in Proposition 1.3(1) .