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

Abstract: A random group in the triangular binomial model $\Gamma(n,p)$ is given by the presentation $\langle S|R \rangle$, where $S$ is a set of $n$ generators and $R$ is a random set of cyclically reduced relators of length 3 over $S$, with each relator included in $R$ independently with probability $p$. When $n\rightarrow\infty$, the asymptotic properties of groups in $\Gamma(n,p)$ vary widely with the choice of $p=p(n)$. By Antoniuk-Łuczak-Świątkowski and Żuk, there exist constants $C, C'$, such that a random triangular group is asymptotically almost surely (a.a.s.) free, if $p < Cn^{-2}$, and a.a.s. infinite, hyperbolic, but not free, if $p\in (C'n^{-2}, n^{-3/2-\varepsilon})$. We generalize the second statement by finding a constant $c$ such that, if $p\in(cn^{-2}, n^{-3/2-\varepsilon})$, then a random triangular group is a.a.s. not left-orderable. We prove this by linking left-orderability of $\Gamma \in \Gamma(n,p)$ to the satisfiability of a random propositional formula, constructed from the presentation of $\Gamma$. The left-orderability of quotients will be also discussed.

group theorygeometric topologymetric geometry

Audience: researchers in the topic

McGill geometric group theory seminar