Random groups at density $$d<1/2$$: sharp length inequalities for generalized torsion and a fixed-width exclusion via first-order transfer

Let $$G=G(m,d,L)$$ G = G ( m , d , L ) be a random group in Gromov’s density model at density d<1/2 d < 1 2 . We prove a sharp quantitative constraint on products of conjugates equal to the identity: for every ε >0 ε > 0 , with overwhelming probability as L→ ∞ L → ∞ , every n≥ 1 n ≥ 1 , and every relation represented by freely reduced words W₀=∏ ᵢ₌₁ⁿ hᵢ⁻¹ g hᵢ =1 in G W 0 = ∏ i = 1 n h i - 1 g h i = 1 in G with g nonempty satisfies the inequality ∑ ᵢ₌₁ⁿ |hᵢ| \;>\; 1-2d-ε /2\,L \;-\; n/2\,|g|. ∑ i = 1 n | h i | > 1 - 2 d - ε 2 L - n 2 | g | . Here all lengths are word lengths. The proof is a short van Kampen diagram argument: Ollivier’s sharp isoperimetric inequality forces a 2-cell contributing a large portion of its boundary to the outer boundary, and a count of the surviving boundary letters according to their origin yields the lower bound. As consequences, we obtain uniform short-witness exclusions and width–length tradeoffs for generalized torsion at every density d<1/2 d < 1 2