Random Groups at Density $d<1/2$: Sharp Length Inequalities for Generalized Torsion and a Fixed-width Exclusion via First-order Transfer
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910008854183936 |
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| author | Baik, Hyungryul |
| author_facet | Baik, Hyungryul |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_02183 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random Groups at Density $d<1/2$: Sharp Length Inequalities for Generalized Torsion and a Fixed-width Exclusion via First-order Transfer Baik, Hyungryul Group Theory Geometric Topology 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. |
| title | Random Groups at Density $d<1/2$: Sharp Length Inequalities for Generalized Torsion and a Fixed-width Exclusion via First-order Transfer |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2602.02183 |