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

We show that a random group $\mathrm{\Gamma }$ in the triangular binomial model $\mathrm{\Gamma }(n,p)$ is a.a.s. not left-orderable for $p\in(cn^{-2}, n^{-3/2-\varepsilon})$,where$c, \varepsilon$areanyconstantssatisfying$\varepsilon>0$,${c>(1/8)\log_{4/3}{2}\approx 0.3012}$\mathrm{.}Wealsoprovethatif$p\geq (1+\varepsilon)(\log n)n^{-2}$foranyfixed$\varepsilon>0$, then a random $\mathrm{\Gamma }\in \mathrm{\Gamma }(n,p)$ 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.