Sobhi Massalha | Searcharxiv

arXiv2024

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 Σ, all the solutions of Σover a random group of density d<\frac{1}{2} are projected from solutions of Σover 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.

math.GR↗

arXiv2024

Sentences over Random Groups II: Sentences of Minimal Rank

Random groups of density d<\frac{1}{2} are infinite hyperbolic, and of density d>\frac{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 F_{k} and over a random group of density d<\frac{1}{2}. We conclude that any given sentence of minimal rank is a truth sentence over the free group F_{k} if and only if it is a truth sentence over random groups of density d<\frac{1}{2}.

math.GR↗