Sentences over Random Groups I: Existential Sentences

Random groups of density d<\frac{1}{2} are infinite hyperbolic, and of density d>\frac{1}{2} are finite. We prove that for any given system of equations \Sigma, all the solutions of \Sigma over a random group of density d<\frac{1}{2} are projected from solutions of \Sigma over the free group F_{k}, with overwhelming probability, where k is the rank of the group. We conclude that any given sentence in the Boolean algebra of universal sentences, is a truth sentence over F_{k} if and only if it is a truth sentence over random groups of density d<\frac{1}{2}, with overwhelming probability.