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arXiv · 2104.09192

Density of random subsets and applications to group theory

Tsung-Hsuan Tsai
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Abstract

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 $0\leq d<\frac{1}{120m^2\ln(2m)}$.

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Tsung-Hsuan Tsai. 2021-10-25. Density of random subsets and applications to group theory. https://doi.org/10.4171/jca%2F63

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