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It is known (and not complicated to prove) that for a finitely generated not virtually nilpotent group $G$, we can pass to a quotient $G/N$ of $G$, such that the quotient is just not virtually nilpotent, i.e. it is not virtually nilpotent, but every proper quotient of it is.

Does a similar argument holds in the broader realm of locally compact groups? that is,

Does any compactly generated locally compact non nilpotent-by-compact group $G$ have a non nilpotent-by-compact quotient $G/N$ such that any proper quotient of $G/N$ is nilpotent-by-compact?

One may assume some relaxing conditions, for example that $G$ is a real connected Lie group.

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    $\begingroup$ In the setting of locally compact groups, it is better to consider the class of groups with polynomial growth, because it is more stable. Indeed the class of nilpotent-by-compact groups is not stable under taking closed subgroups. $\endgroup$
    – YCor
    Jul 14, 2015 at 11:37
  • $\begingroup$ In the case of real connected Lie groups, a result in the spirit of what you're looking at is Lemma 4.1 in msp.org/gt/2008/12-1/gt-v12-n1-p11-s.pdf $\endgroup$
    – YCor
    Jul 14, 2015 at 11:54
  • $\begingroup$ Ha ha, I was just going through a paper of yours looking for clues, thanks @YCor! $\endgroup$
    – Snoop Catt
    Jul 15, 2015 at 7:18
  • $\begingroup$ A small follow up question - in theorem 4.3 of msp.org/gt/2008/12-1/gt-v12-n1-p11-s.pdf, you do not assume the the Lie group is real, but you invoke Lemma 4.1, which is proved for this case only. What am I missing? Thanks! $\endgroup$
    – Snoop Catt
    Jul 19, 2015 at 10:10
  • $\begingroup$ In theorem 4.3 it's written "connected Lie group", so there's little ambiguity that it's a real Lie group. A connected $p$-adic Lie group is necessarily reduced to the trivial group, and a complex Lie group can be viewed as a real Lie group. $\endgroup$
    – YCor
    Jul 19, 2015 at 12:30

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Every compactly generated locally compact group is either of polynomial growth, or it has a quotient that is just non-(polynomial growth). The same also works for a number of similar properties in place of 'polynomial growth', e.g. 'compact-by-nilpotent'. See Theorem 3.12 of this preprint: http://arxiv.org/abs/1509.06593 (it should say "for any compactly generated locally compact group $G$...")

I suspect there are other short proofs of this result using only the published literature, but this is the argument I know of.

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