Sentences over Random Groups I: Existential Sentences

TL;DR

This paper demonstrates that for random groups of density less than 1/2, the solutions to equations mirror those over free groups, establishing a probabilistic equivalence for universal sentences.

Contribution

It proves that solutions of equations over random groups of density less than 1/2 are essentially projections from free group solutions, linking algebraic properties to probabilistic group models.

Findings

Solutions over random groups of density d<1/2 are projections from free group solutions.

Universal sentences are true over random groups if and only if they are true over free groups.

Random groups of density d<1/2 are infinite hyperbolic, similar to free groups.

Abstract

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