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

Abstract Let $$G=G(m,d,L)$$ G = G ( m , d , L ) be a random group in Gromov’s density model at density $$d0$$ ε > 0 , with overwhelming probability as $$L\rightarrow \infty $$ L → ∞ , every $$n\ge 1$$ n ≥ 1 , and every relation represented by freely reduced words $$ W_0=\prod _{i=1}^n h_i^{-1} g h_i =1 \quad \text {in } G $$ W 0 = ∏ i = 1 n h i - 1 g h i = 1 in G with g nonempty satisfies the inequality $$ \sum _{i=1}^n |h_i| \;>\; \frac{1-2d-\varepsilon }{2}\,L \;-\; \frac{n}{2}\,|g|. $$ ∑ 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