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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{...

Suppose we have a finitely presented group $G$ with decidable word problem. Is it decidable to check whether a given element $x\in G$ has finite order or infinite?

Is there any reasonably updated list of (representative) examples of finitely presented groups with undecidable word problem? By "representative" I mean "avoiding obvious redundancy", i.e. examples ...