Sentences over Random Groups II: Sentences of Minimal Rank

TL;DR

This paper establishes a uniform quantifier elimination procedure for minimal rank formulas in free and random groups, linking their logical properties and showing equivalence of truth sentences across these groups.

Contribution

It introduces a quantifier elimination method for minimal rank formulas, connecting the logical theories of free and random groups of density less than 1/2.

Findings

Existence of a quantifier elimination procedure for minimal rank formulas.

Equivalence of truth sentences in free and random groups of density d<1/2.

Minimal rank formulas define the same sets over free and random groups.

Abstract

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 \varphi(p) that belongs to the Boolean algebra of two quantifiers, so that the two formulas V(p) and \varphi(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}.