Some small cancellation properties of random groups

Abstract

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant $1-2d$ depending upon the density parameter $d$. This implies in particular a property generalizing the ordinary $C&#39;$ small cancellation condition, which could be termed ``macroscopic small cancellation&#39;&#39;. This also sharpens the evaluation of the hyperbolicity constant $δ$. As a consequence we get that the standard presentation of a random group at density $d<1/5$ satisfies the Dehn algorithm and Greendlinger&#39;s Lemma, and that it does not for $d>1/5$.