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.
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, 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, 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 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.
Yuhasz approach was to prove that every one-relator group satisfies certain version of small cancelation condition and hence has solvable conjugacy problem. Of course if the relator satisfies $C'(1/6)$ the group is hyperbolic and we are done. I think from what he proved the CAT(0) conditions $C(3)-T(6)$ and $C(4)-T(4)$ are Ok too but I do not think all cases are covered in his paper. It was long since I looked at his text, so I may forget something. Victor Guba knows the text well.
