Density of random subsets and applications to group theory

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 $densA:={\log }_{|E|}(|A|)$. 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 $dens(A\cap B)=densA+densB-1$. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density $\lambda /2$ for any $0<\lambda <1$, characterizing the $C^{\prime }(\lambda )$-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol'shanskii from density $0$ to density $0\le d<\frac{1}{120m^2\ln (2m)}$.

version 2, 41 pages

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