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 $C'$ small cancellation condition, which could be termed ``macroscopic small cancellation''. This also sharpens the evaluation of the hyperbolicity constant $\delta$. As a consequence we get that the standard presentation of a random group at density $d<1/5$ satisfies the Dehn algorithm and Greendlinger's Lemma, and that it does not for $d>1/5$.