Density of random subsets and applications to group theory
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916912330440704 |
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| author | Tsai, Tsung-Hsuan |
| author_facet | Tsai, Tsung-Hsuan |
| contents | Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a precise meaning of Gromov's \textit{intersection formula}: "Random subsets" $A$ and $B$ of a finite set $E$ satisfy $dens (A\cap B) = dens A + dens B -1$.
As an application, we exhibit a phase transition phenomenon for random presentations of groups at density $λ/2$ for any $0<λ<1$, characterizing the $C'(λ)$-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol'shanskii from density $0$ to density $0\leq d<\frac{1}{120m^2\ln(2m)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_09192 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Density of random subsets and applications to group theory Tsai, Tsung-Hsuan Group Theory Probability 2020: 20F05, 20F06, 60C05 Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a precise meaning of Gromov's \textit{intersection formula}: "Random subsets" $A$ and $B$ of a finite set $E$ satisfy $dens (A\cap B) = dens A + dens B -1$. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density $λ/2$ for any $0<λ<1$, characterizing the $C'(λ)$-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol'shanskii from density $0$ to density $0\leq d<\frac{1}{120m^2\ln(2m)}$. |
| title | Density of random subsets and applications to group theory |
| topic | Group Theory Probability 2020: 20F05, 20F06, 60C05 |
| url | https://arxiv.org/abs/2104.09192 |