Non-orderability of random triangular groups by using random 3CNF formulas

TL;DR

This paper demonstrates that random triangular groups are almost surely not left-orderable within certain probability ranges, using 3CNF formulas to encode and prove conditions for non-orderability.

Contribution

It introduces a novel approach of encoding left-orderability conditions into 3CNF formulas and proves their unsatisfiability in random groups, establishing non-orderability results.

Findings

Random groups in the specified model are almost surely not left-orderable for certain p ranges.

Random groups have no non-trivial left-orderable quotients when p exceeds a threshold.

The method of encoding orderability conditions into 3CNF formulas is effective for probabilistic proofs.

Abstract

We show that a random group $\Gamma$ in the triangular binomial model $\Gamma(n, p)$ is a.a.s. not left-orderable for $p\in(cn^{-2}, n^{-3/2-\varepsilon})$, where $c, \varepsilon$ are any constants satisfying $\varepsilon>0$, ${c>(1/8)\log_{4/3}{2}\approx 0.3012}$. We also prove that if $p\geq (1+\varepsilon)(\log n)n^{-2}$ for any fixed $\varepsilon>0$, then a random $\Gamma\in \Gamma(n,p)$ has a.a.s. no non-trivial left-orderable quotients. We proceed by constructing 3CNF formulas, which encode necessary conditions for left-orderability and then proving their unsatisfiability a.a.s.