Critical densities for random quotients of hyperbolic groups

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various models. To cite this article: Y. Ollivier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).