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I'm curious about finitely generated subgroups in the isometry group of Euclidean spaces. I know the isometry group is the semi-direct product of translation groups with orthogonal groups. Also Bieberbach theorems characterize virtually abelian subgroups acting on cocompactly and properly on Euclidean spaces. Except those, I want to know an answer to the following questions:

  1. Does there exist non-(virtually)-abelian solvable subgroups in $\mathrm{Isom}(\mathbb{E}_n)$? Non-(virtually)-abelian nilpotent groups?

  2. Any general results about non-discrete finitely generated subgroups?

Thanks a lot!

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  • $\begingroup$ The question is quite trivial since all finite groups appear. It would make more sense with "virtually". The short answer is that (a) discrete virtually solvable subgroups are virtually abelian (b) virtually nilpotent subgroups are virtually abelian (c) there exist virtually solvable subgroups that are not virtually abelian [hence, by a,b, neither discrete nor virtually nilpotent] $\endgroup$
    – YCor
    Apr 4, 2021 at 9:46
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    $\begingroup$ @YCor, yes, I wanted "non-virtually abelian" and will edit it. But could you provide me some references to your answers? in particular, relevant to (b)? I tried to search in literatures but found nothing. Thanks a lot! $\endgroup$
    – stephen
    Apr 4, 2021 at 9:54

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