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I'm looking for results of the form "every infinite solvable group contains <...> as a subgroup". Specifically, I believe:

If $G$ is infinite solvable, finitely generated and not virtually cyclic, then $G$ contains as subgroup either the metabelian group $\mathbb Z[1/mn]\rtimes\mathbb Z$, with $\mathbb Z$ acting by multiplication by $m/n$ [possibly $m=n=1$], or the wreath product $\mathbb Z/p\wr\mathbb Z$ for some prime $p$.

(I know that every infinite solvable group contains one of these two subgroups as a section [= quotient of two subgroups], by Kropholler's 1984 result; but really need them as subgroups. In fact, if the answer to the question is "no", I'm very much interested in a corrected statement with as few extra cases as possible.)

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  • $\begingroup$ Are you missing some conditions in 1? Finitely generated? Non-abelian? $\endgroup$
    – Jeremy Rickard
    Nov 2, 2020 at 9:52
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    $\begingroup$ Ouch, yes! I'll edit the question $\endgroup$
    – grok
    Nov 2, 2020 at 9:58
  • $\begingroup$ Both questions are non-trivial and I don't expect an answer to any of them to solve the other one: I'd suggest to post the second one separately. $\endgroup$
    – YCor
    Nov 2, 2020 at 10:13
  • $\begingroup$ Clarified $m=n=1$ allowed. Hopefully "2." is useful in understanding "1."; one could weaken "2." in requiring $A$ to have infinite index in its normalizer, and then I think it's really an intermediate step towards "1.". $\endgroup$
    – grok
    Nov 2, 2020 at 11:12
  • $\begingroup$ Anyway I'm close to an answer to 1 and it won't answer 2, so I'd be reluctant to post a long answer complete answer to 1 if it doesn't answer the whole question. $\endgroup$
    – YCor
    Nov 2, 2020 at 11:14

1 Answer 1

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It's not true: the lamplighter group $L=\mathbb Z/p\wr\mathbb Z$ has a universal central extension (by $H_2(L,\mathbb Z)=(\mathbb Z/p)^\infty$) which does not contain any lamplighter group as a subgroup.

Thanks to @YCor for offsite discussions on this.

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    $\begingroup$ I'm not sure it's universal in the usual sense (I think a group has a universal central extension iff it's perfect). Anyway you might want to include explicitly the extension. $\endgroup$
    – YCor
    Nov 18, 2020 at 21:03

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