STRUCTURAL PROPERTIES OF REDUCED C*-ALGEBRAS ASSOCIATED WITH HIGHER-RANK LATTICES

Theorem 1.2, as stated in the paper, asserts that for any r > 0 and any n ∈ N there exist γ1,…,γn ∈ Γ with |γi| ≤ C r (C independent of n) such that w(γ1,…,γn) ≠ e for every nontrivial w ∈ Γ * F_n whose Γ-coefficients have S-length ≤ r . Taking w = x_i x_j^{-1} (which has no Γ-coefficients and hence is permitted) forces γ_i ≠ γ_j for all i ≠ j; thus at least n distinct elements are required in the ball B_Γ(S, C r), which is finite (indeed |B_Γ(r)| ≤ e^{α r} for some α > 0) . For fixed r and arbitrarily large n this is impossible. Moreover, the actual proof provided in the paper constructs a single r-free element γ of length O(r) (n = 1 case) using the geometric function ψ_r, its Lipschitz control L_r ≤ e^{4r}, and a mixing argument, culminating in Lemma 4.5 and Proposition 6.3 . It does not supply the simultaneous construction for general n. The model correctly identifies the counting obstruction and proposes the corrected bound |γ_i| ≤ C1 r + C2 log n together with a plausible ping–pong-with-coefficients construction to achieve it.