Random Groups at Density $d<1/2$: Sharp Length Inequalities for Generalized Torsion and a Fixed-width Exclusion via First-order Transfer

Abstract

Let $G$ be a random group in Gromov's density model $G(m,d,L)$ with $d<\tfrac12$. We prove a sharp quantitative constraint on products of conjugates equal to the identity: for every $n\ge1$ and $\varepsilon>0$, with overwhelming probability as $L\to\infty$, any tight word \[ W=\prod_{i=1}^n h_i^{-1} g h_i =1 \quad\text{in } G \] (with $g\neq 1$ as a word) satisfies the inequality \[ \sum_{i=1}^n \len{h_i} \;>\; \frac{1-2d-\varepsilon}{2}\,L \;-\; \frac{n}{2}\,\len{g}. \] 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 simple boundary block-counting estimate yields this corridor-type lower bound. As consequences we obtain uniform short-witness exclusions and width--length tradeoffs for generalized torsion at every density $d<\tfrac12$. We also deduce that random groups have no generalized torsion of any fixed width as a corollary of the recent first-order transfer theorem of Kharlampovich, Miasnikov, and Sklinos.