# Are braid groups conjugacy separable?

I would like to re-ask a question that was raised in the comments here: Normal subgroups of braid groups

Namely, a group $G$ is called conjugacy separable, if it holds that for every two elements $x,y\in G$ which are not conjugate in $G$, there exists a finite index normal subgroup $N$ such that $xN$ and $yN$ are not conjugate in $G/N$.

Is it known if the braid groups $B_n$ are conjugacy separable?

• For the 3-strand braid group, which is isomorphic to the fundamental group of the trefoil knot complement, a Seifert-fibered manifold with boundary, conjugacy separability was proved e.g. here: ams.org/mathscinet-getitem?mr=1980432 – Ian Agol Apr 23 '16 at 3:55

As far as I know, nothing has been published on the conjugacy separability of $B_n$ for $n\geq 4$. Quite a lot of recent effort has been expended trying to work out whether or not braid groups are virtually compact special. If that were true, then conjugacy separability would follow (at least for a finite-index subgroup) by Minasyan's theorem. Unfortunately, the wonderful recent work of Haettel seems to indicate that braid groups are probably not virtually compact special.