Let $G$ be a discrete and finitely generated group. Recall that $\{F_n\}_{n \in \mathbb{N}}$ is a *Følner sequence* if $|g F_n \cup F_n|/|F_n| \rightarrow 1$ for every $g \in G$. As is well known, existence of a Følner sequence is equivalent to amenability of $G$.

It is often said that Følner sequences have *strange* shapes. My soft question is: which examples do we have that support this claim? Of course, if $G$ is of subexponential growth then a subsequence of balls forms a Følner sequence, and this does not have a *weird* shape. Hence, more specifically: which examples of groups of exponential growth do we know that have explicit Følner sequences not made of balls?

As instances of the examples I am asking for, <https://mathoverflow.net/questions/143135/star-shaped-folner-sequence> asks for Følner sets of a certain form, while an answer of <https://mathoverflow.net/questions/148995/folner-sets-and-balls> gives explicit sequences made of *rectangles* (as opposed to balls). Likewise, the ***ax + b*** group has a Følner sequence made of rectangles where one side is exponentially larger than the other.