Density of random subsets and applications to group theory

Abstract

Developing an idea of M. Gromov (1993), we study the intersection formula for random subsets with density. The density of a subset $A$ in a finite set $E$ is defined by $\mathrm{dens}A:={\log }_{|E|}(|A|)$. The aim of this article is to give a precise meaning of Gromov’s intersection formula: “Random subsets” $A$ and $B$ of a finite set $E$ satisfy $\mathrm{dens}(A\cap B)=\mathrm{dens}A+\mathrm{dens}B-1$.

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 (1996) from density $0$ to density $0\le d<1/(120m^2\ln (2m))$.