2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66278We 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.91 pages ; 3rd version: improved redaction, corrected typosGroup TheoryProbability20P05 (Primary) 20F06, 20F67, 60B99 (Secondary)text