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If $H$ is a finitely generated subgroup of $G$ and if $H$ given by say a finite set of words which generate it, then the generalized word problem for $H$ in $G$ is the problem of deciding for an arbitrary word $w$ in $G$ whether or not $w$ lies in the subgroup $H$.

In "Computable algebra and group embeddings", it is shown that one can solve the generalized word problem for finitely generated, abelian-by-nilpotent groups. Is this result generalized to the finitely generated abelian by polycyclic groups?

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    $\begingroup$ Context: $G$ is a group with a given finite generating family (so that "words" makes sense). To say that the generalized word problem is solvable in some class $C$ of f.g. groups means that for every $G\in C$ and every finite generating family (a given one is enough), and every f.g. subgroup $H\subset G$ given by a finite generating set of words, the generalized word problem for $(G,H)$ is solvable (I still have a doubt whether the algorithm should be uniform in $H$, which would sound a more natural assumption). $\endgroup$
    – YCor
    Commented Mar 3, 2016 at 12:59
  • $\begingroup$ I am struggling to understand "finitely generated abelian by polycyclic groups". You must surely be aware that, for classes A and B of groups "A by B groups" can have one of two different meanings, so could you say exacly what you mean? $\endgroup$
    – Derek Holt
    Commented Mar 3, 2016 at 14:32
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    $\begingroup$ According to a survey article by C.F. Miller written in about 1989, this is unknown. Of course it is possible that there has been further progrss since then. $\endgroup$
    – Derek Holt
    Commented Mar 3, 2016 at 14:39
  • $\begingroup$ @DerekHolt Very likely it means abelian kernel, polycyclic quotient as it's an interesting class of groups (according to the same convention kernel-by-quotient, a polycyclic-by-abelian f.g. group is f.g., hence polycyclic, so it would be stupid to refer to this class) $\endgroup$
    – YCor
    Commented Mar 3, 2016 at 14:48
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    $\begingroup$ by an abelian by polycyclic group G, I mean a group with an abelian normal subgroup N such that G/N is polycyclic $\endgroup$
    – user182085
    Commented Mar 3, 2016 at 14:49

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