Recall that a group $G$ satisfies max (or is said to be Noetherian) if all its proper subgroups are finitely generated. Similarly $G$ satiesfies 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. Noetheless 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.