Some Small Cancellation Properties of Random Groups

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 cancellation”. 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’s Lemma, and that it does not for d > 1/5. For this we establish a version of the local-global principle for hyperbolic spaces (Cartan-Hadamard-Gromov theorem) involving arbitrarily small loss in the isoperimetric constant.