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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?

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  • $\begingroup$ 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 $\endgroup$ – Ian Agol Apr 23 '16 at 3:55
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(26 April 2016: Updated to give a fuller answer.)

I'm fairly confident this question is still open.

As Ian Agol points out in comments, the 3-strand braid group is, by a happy accident, also the fundamental group of the trefoil-knot complement, which has been known to be conjugacy separable for a long time. (All compact, orientable 3-manifold groups are now known to be conjugacy separable.)

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.

One can certainly imagine attempting a proof by induction, using the Birman exact sequence. But there are quite a few technical difficulties to overcome, so making an argument like that work would be a great achievement.

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