In a [previous question](https://mathoverflow.net/questions/425966/examples-of-locally-hyperbolic-groups), I asked about hyperbolic groups in which every finitely generated subgroup is hyperbolic. I am now curious about the reverse question: what are some examples of hyperbolic groups containing finitely generated non-hyperbolic groups?


Any non-coherent hyperbolic group will give an example (e.g. via the Rips construction). Do there exist examples of coherent hyperbolic groups containing f.g. non-hyperbolic subgroups?



Examples of hyperbolic groups with non-hyperbolic subgroups
It is an open problem to find a coherent hyperbolic group with a finitely generated, non-hyperbolic subgroup. See [Wise's survey article](https://www.degruyter.com/document/doi/10.1515/9780691185897-014/html?lang=en) for the state of the art on coherent groups.


Wise, Daniel T. (3-MGL)
**An invitation to coherent groups.**
*What's next?—the mathematical legacy of William P. Thurston*, 326–414,
Ann. of Math. Stud., 205, *Princeton Univ. Press, Princeton, NJ, 2020*.


I suspect this is listed as a question there, but if not, it's very much in that spirit.


The theorem of Gersten that I mentioned [in answer to your last question](https://mathoverflow.net/a/425992/1463) showed that there is no 2-dimensional example. All known coherent 3-dimensional groups are essentially 3-manifold groups. [Sam Nead's answer to that question](https://mathoverflow.net/a/425967/1463) explained why finitely generated subgroups of hyperbolic 3-manifold groups are also hyperbolic.


As Matt Zaremsky mentions in comments, the key examples of finitely presented non-hyperbolic subgroups of hyperbolic groups are given by [the original example of Noel Brady](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024610799007644) and the recent work of [Italiano--Martelli--Migliorini](https://arxiv.org/abs/2105.14795). There are also related constructions by [Fujiwara](https://arxiv.org/abs/2106.08549), [Groves--Manning](https://arxiv.org/abs/2205.11290), [Kropholler](https://arxiv.org/abs/1808.09505), [Llosa Isenrich--Martelli--Py](https://arxiv.org/abs/2112.06531) and [Lodha](https://arxiv.org/abs/1403.6716).


It would be very surprising if any of these examples were to turn out to be coherent. In fact, it is reasonable to conjecture that these examples are all incoherent, but I believe it is unknown in each case. Proving that Brady's example (say) is incoherent wouldn't answer your question, but would be a nice thing to do.


