Sentences over Random Groups I: Existential Sentences

Random groups of density d<1/2 are infinite hyperbolic, and of density d>1/2 are finite. We prove that for any given system of equations Σ, all the solutions of Σ over a random group of density d<1/2 are projected from solutions of Σ over the free group Fₖ, 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 Fₖ if and only if it is a truth sentence over random groups of density d<1/2, with overwhelming probability.