First-order sentences in random groups III

TL;DR

This paper proves that in random groups with density less than 1/2, the truth of first-order sentences aligns with their truth in non-abelian free groups, revealing a deep logical equivalence.

Contribution

It establishes a precise correspondence between first-order logical properties in random groups and non-abelian free groups for densities below 1/2.

Findings

First-order sentences are almost surely true in random groups at density d<1/2 if and only if true in free groups.

The result links logical properties of random groups to classical free groups.

It advances understanding of the logical structure of random groups in geometric group theory.

Abstract

We show that a first-order sentence is almost surely true in a random group of density d<1/2 if and only if it is true in a non-abelian free group.