The braid group on 3 strands has the presentation $\langle x,y \;|\; xyx=yxy\rangle$. A group $G$ is called right-orderable if there is a total order $<$ on the set $G$ such that if $a<b$ then $ac<bc$ for all $c\in G$.
It is known that braid groups are right-orderable.


Is there a non-right-orderable torsion-free quotient group of the braid group on 3 strands?



Is there a non-right-orderable torsion-free quotient group of the braid group on 3 strands?
Yes, there are many such examples. The braid group is isomorphic to the fundamental group of the trefoil knot complement. The trefoil knot $T$ admits many Dehn fillings, parameterized by $r\in \mathbb{Q} \cup \{\infty\}$. If $|r|\geq 1$, then the Dehn filling $S^3\_r(T)$ is an L-space. Moreover, it is usually Seifert-fibered and has torsion-free fundamental group. By Theorem 4 of [Boyer-Gordon-Watson](http://www.cirget.uqam.ca/boyer/Publications/bgw.pdf), these Dehn fillings do not have orderable fundamental group (the notion of L-space is actually not relevant to the answer to this question, it just gave the quickest way to cite the literature). 


