I am aware that the Spelling Theorem of B. B. Newman implies that one-relator groups with torsion are hyperbolic, and thus have a solvable conjugacy problem. My understanding is that for one-relator groups without torsion, the conjugacy problem is still open, though the most recent reference I have for this is over 20 years old.


Has there been any recent developments in this area? Any references for work in this area would be appreciated. 


Thanks.



Reference request: Recent progress on the conjugacy problem for torsion-free one-relator groups?
As mentioned in the comments, this is still considered an open problem. I thought I'd flesh out a few aspects. A solution was claimed in 1992 by [Juhasz](https://link.springer.com/chapter/10.1007%2F978-1-4613-9730-4_3), but it seems to have failed to convince experts. The small cancellation theory involved in the proof seems to be very intricate, though I am not aware of any concrete example of a gap in the proof. Perhaps someone can fill me in on this.


At the Winter One-relator Workshop at the University of East Anglia two years ago, in 2018, (see [WOW](https://archive.uea.ac.uk/%7Efga12juu/WOW.html), this very question was brought up. I was not there myself, but it was organised by my supervisor; I myself asked him this question not long ago, and he conveyed that the consensus was that the problem remained open.


As another example from the literature, in [these excellent notes](https://www3.nd.edu/%7Eandyp/notes/OneRelator.pdf) by Andrew Putman, one reads "Whether or not torsion-free one-relator groups have a solvable conjugacy problem is a famous and difficult open question". I am not sure exactly when these notes are from, but certainly after 2016. So: the problem remains open.


