TITLE: Is being Noetherian a quasi-isometry invariant for f.g. groups?
QUESTION [5 upvotes]: 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.

REPLY [8 votes]: 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.