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.**