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*.**



Is being Noetherian a quasi-isometry invariant for f.g. groups?
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](http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/probFP.html), 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](http://www.math.cornell.edu/~vogtmann/MSRI/Eskin.pdf).)


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.


