Random groups are not left-orderable

TL;DR

This paper demonstrates that in the Gromov density model, random groups almost surely lack non-trivial left-orderable quotients, especially at densities below 1/2, highlighting a significant property of such groups.

Contribution

It establishes that random groups in the Gromov density model are almost surely not left-orderable at any density, extending understanding of their algebraic properties.

Findings

Random groups at any density lack non-trivial left-orderable quotients.

At densities less than 1/2, random groups are not left-orderable.

Overwhelming probability results for properties of random groups.

Abstract

We prove that random groups in the Gromov density model at any density $d$ have with overwhelming probability no non-trivial left-orderable quotients. In particular, random groups at densities $d<\frac{1}{2}$ are not left-orderable.