Non-uniform higher-rank lattices are character rigid

The candidate reproduces the first proof of Theorem 1.2 almost verbatim: reduce to an S‑arithmetic model via arithmeticity and commensurability invariance, then in the rank‑1 factor use Bruhat to write g = w m u and compute [g, a^n] = a^{2n} v_n, appeal to the mixing/decay on split tori (uniform in unipotents) to get lim_n sup_u |τ(a^{2(m−n)}u)|=0, and conclude τ(g)=0 by the generalized Bekka lemma; finite‑dimensional characters are handled separately. These steps match the paper’s statement and proof: statement of character rigidity in the rank‑1 factor setting (Theorem 1.2), the arithmetic reduction (Proposition 3.5), the mixing/decay theorem on split tori (Theorem 6.1), the explicit commutator calculation in §7.1, and the generalized Bekka lemma (Lemma 2.12). The minor differences are: (i) the candidate occasionally writes h_n = u_- a^n whereas the paper takes x_n = a^n; (ii) the candidate quotes a slightly stronger hypothesis for the Bekka lemma than is needed; and (iii) the candidate doesn’t explicitly note that an infinite‑dimensional character is weakly mixing (which justifies invoking Theorem 6.1), though this follows from the paper’s weakly mixing vs. finite‑dimensional dichotomy for characters. The characteristic ≠ 2 hypothesis is also correctly tied to the Raghunathan–Venkataramana input (Theorem 6.2), which the paper identifies as the only place where char(F) ≠ 2 is used. Overall, the proofs align in structure and ingredients. See Theorem 1.2, Proposition 3.5, Theorem 6.1, Lemma 2.12, and the first proof of Theorem 1.2 in §7.1 for the corresponding steps in the paper .