Recall that a group $G$ satisfies max (or is said to be Noetherian) if all its proper subgroups are finitely generated. Similarly $G$ satisfies max-n if all its normal subgroups are normal closures of finite subsets. Note that property max is "closed with respect to extension", i.e if $N \unlhd G$ and $G \backslash N$ have this property than so does $G$ (in particular this implies that polycyclic groups satisfy max). However max-n is not inherited by subgroups. Nonetheless we have the following Theorem:

If a group $G$ satisfies min-n (resp. max-n) and $H$ is a subgroup of $G$ with finite index, then $H$ satisfies min-n (resp. max-n).

My question is, if these properties max, min, max-n and min-n are preserved under quasi-isometries (of f.g. groups).

Edit 1: as YCor pointed out in a comment the answer is negative for max-n and min-n. The question for max and min seems still open (in general), however we may have a lack of examples.

In this sense we may collect examples of groups satisfying max resp. min.

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    $\begingroup$ Not max-n: $F_2\times F_2$ being quasi-isometric to irreducible lattices in $SL_2(\mathbf{Q}_p)^2$, which are just-infinite hence max-n. They are also QI to Burger-Mozes groups which are simple and hence min-n, so min-n is also not QI-invariant. For max, it's probably open (too few known examples). For min, I guess there are too few known examples as well. $\endgroup$ – YCor Sep 9 '15 at 9:10
  • $\begingroup$ This is many years later, but the property "Max" for Noetherian surely ought to be "Emmy" :-) $\endgroup$ – Carl-Fredrik Nyberg Brodda Feb 16 at 8:38

For finitely presented groups, you're close to two well known open questions/conjectures.

Question 1: If a finitely presented group is Noetherian, is it virtually polycyclic?

(This question is FP11 here, where it's attributed to S. Ivanov.)

Question 2: The class of virtually polycyclic groups is closed under quasi-isometry.

(Alex Eskin conjectures this here.)

Putting these together, I think it's fair to say that, for finitely presented groups, it's unknown whether the property of being Noetherian is a qi-invariant.


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