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I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation $\langle X,R\rangle$ for a Noetherian group, \begin{equation*} \begin{array}{ll} \text{Input:} & u,v_1,\ldots,v_n \text{ words in } X^{\pm}\\ \text{Decide:}& u \in \langle v_1,\ldots,v_n \rangle \end{array} \end{equation*}

Of course, relevant to this is the possibility that every finitely presented Noetherian group is virtually polycyclic (and so has solvable MP). Is this still an open conjecture question? Any information on decision problems for Noetherian groups will be also greatly appreciated.

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    $\begingroup$ I don't know if it's a conjecture (the claim that every f.p. noetherian group is virtually polycyclic); it's an open question. You conjecture something when you strongly believe it's true. $\endgroup$ Commented Nov 13, 2018 at 17:26
  • $\begingroup$ @Ycor agreed! So, is it still open? $\endgroup$ Commented Nov 13, 2018 at 17:28
  • $\begingroup$ I have just discovered that the alluded problem still appears in the new (2018) version of the "Kourovka Notebook" (Collection of unsolved problems in group theory). Concretely: 11.38. Does there exist a finitely presented Noetherian group which is not almost polycyclic? (S. V. Ivanov) See kourovka-notebook.org $\endgroup$ Commented Nov 13, 2018 at 22:37
  • $\begingroup$ For noetherian (not f.p.) groups, solvable WP (i.e. solvable membership for the trivial subgroup) can be undecidable. I don't know whether solvable MP holds for noetherian groups with solvable WP. $\endgroup$ Commented Nov 13, 2018 at 23:10
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    $\begingroup$ It's certainly still an open problem. $\endgroup$ Commented Nov 15, 2018 at 18:05

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