# Subgroup membership problem for Noetherian groups

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.

• 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. – YCor Nov 13 '18 at 17:26
• @Ycor agreed! So, is it still open? – suitangi Nov 13 '18 at 17:28
• 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 – suitangi Nov 13 '18 at 22:37
• 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. – YCor Nov 13 '18 at 23:10
• It's certainly still an open problem. – HJRW Nov 15 '18 at 18:05