密度d<1/2下的随机群：关于广义 torsion 的精确长度不等式及通过一阶转移的固定宽度排除
Random Groups at Density $d<1/2$: Sharp Length Inequalities for Generalized Torsion and a Fixed-width Exclusion via First-order Transfer
AI总结：
该研究在密度d<1/2下证明了随机群中广义torsion的精确长度不等式，并通过一阶转移定理得出无固定宽度的广义torsion结论。
摘要：
设G为在Gromov密度模型G(m,d,L)中的随机群，其中d<1/2。我们证明了关于共轭乘积等于单位元的精确量化约束：对于每个n≥1和ε>0，当L趋向于无穷大时，任何紧致词W=prod_{i=1}^n h_i^{-1}g h_i=1在G中（其中g≠1作为词）满足不等式sum_{i=1}^n len(h_i) > (1-2d-ε)/2 * L - n/2 * len(g)。证明是一个简短的van Kampen图论论证：Ollivier的精确等周不等式迫使一个2细胞贡献其大部分边界到外边界，而一个简单的边界块计数估计得出这种走廊型下界。作为结果，我们获得在任意密度d<1/2下的统一短见证排除和广义torsion的宽度-长度权衡。我们还得出随机群没有任何固定宽度的广义torsion，这是Kharlampovich、Miasnikov和Sklinos最近的一阶转移定理的推论。
英文摘要：
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.
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