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.**](http://dx.doi.org/10.1016/j.jalgebra.2012.04.014), J. Algebra 364, 148-154 (2012). [ZBL1261.20034](https://zbmath.org/?q=an:1261.20034).



Examples of "natural" finitely generated groups with an undecidable conjugacy problem
Chuck Miller in [Miller, Charles F., III *On group-theoretic decision problems and their classification*. Annals of Mathematics Studies, No. 68. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971] proves the following two rather nice and natural examples.


**Theorem III.10.** The free product of two free groups with finitely generated amalgamation can have unsolvable conjugacy problem. Further, the finitely presented HNN extension of a free group can have unsolvable conjugacy problem.


(Note that Miller calls HNN extensions 'Strong Britton extensions').


Now by Bass-Serre theory, there is a natural action of an amalgamated free product/HNN on the associated Bass-Serre tree, which should satisfy your "natural action" criterion.




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Edit: The result mentioned by YCor can also be found in Miller's book.


**Theorem III.23** The group $F\_2 \times F\_2$ has a finitely generated subgroup with undecidable conjugacy problem.


An important side remark, however, is that $F\_2 \times F\_2$ itself has decidable conjugacy problem, as do all RAAGs, in linear time. See [Crisp, John; Godelle, Eddy; Wiest, Bert; *The conjugacy problem in subgroups of right-angled Artin groups.* J. Topol. 2 (2009), no. 3, 442–460.].


