POLYNOMIAL GROWTH, COMPARISON, AND THE SMALL BOUNDARY PROPERTY

The uploaded paper (Naryshkin, 2108.04670) proves Theorem A: every free minimal action of a finitely generated group of polynomial growth has comparison, by first establishing weak (2^{ord}−1)-comparison for free actions (Theorem 3.1) and then using the equivalence of weak m-comparison and comparison in the minimal case (Lemma 2.3). The proof hinges on density bounds along Følner balls and a carefully controlled greedy algorithm on BN×{1,…,m} that maintains a key inequality (3.2) and culminates in subequivalence, then comparison in the minimal setting . The candidate solution reaches the same final claim but its argument contains critical flaws: (i) it assumes a uniform measure gap δ := infμ(μ(B)−μ(A))>0 for open A,B without justification (the difference of two l.s.c. functions need not attain a positive minimum); (ii) it mis-cites a “greedy matching lemma” as Naryshkin’s Lemma 3.1 and imposes a strong pointwise ratio N_F(C, x) ≤ (1/(m+1)) N_F(Bε, x) that is neither proved nor used by the paper; (iii) the step that tries to upgrade a mild surplus δ|B_n| into a uniform multiplicative domination by (d+1) is unsupported and generally false; and (iv) the claimed optimal layering m=d contradicts the paper’s explicit choice m=2^{ord}−1 derived from ball-doubling in groups of polynomial growth. By contrast, the paper’s proof is internally coherent and complete, relying on density identities and a well-specified greedy construction .