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Random groups of density d12 are finite. We prove that for any given system of equations, all the solutions of over a random group of density d<12 are projected from solutions of over the free group F₊, 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₊ if and only if it is a truth sentence over random groups of density d<12, with overwhelming probability.
Sobhi Massalha (Fri,) studied this question.