I am looking for natural groups with undecidable conjugacy problem. By natural, I mean that the word problem should be decidable, and the group should be given by some natural action. I know that $\mathbb{Z}^d \rtimes F_m$ (with a suitable action of $F_m$) has undecidable conjugacy problem. That's very nice, but I'd like to know other examples. I do not care about finite presentation, and I'm also fine with the group being a f.g. *subgroup* of something natural and geometric, which maybe simplifies things. A concrete case I was not able to resolve is whether all f.g. subgroups of right-angled Artin groups have decidable conjugacy problem.

*Šunić, Zoran; Ventura, Enric*, **The conjugacy problem in automaton groups is not solvable.**, J. Algebra 364, 148-154 (2012). ZBL1261.20034.

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