Sentences over Random Groups II: Sentences of Minimal Rank

Random groups of density d<1/2 are infinite hyperbolic, and of density d>1/2 are finite. We prove the existence of a uniform quantifier elimination procedure for formulas of minimal rank (probably the superstable part of the theory). Namely, given a minimal rank formula V(p), we prove the existence of a formula φ(p) that belongs to the Boolean algebra of two quantifiers, so that the two formulas V(p) and φ(p) define the same set over the free group Fₖ and over a random group of density d<1/2. We conclude that any given sentence of minimal rank is a truth sentence over the free group Fₖ if and only if it is a truth sentence over random groups of density d<1/2.