Recall that a discrete group $G$ is amenable if and only if it has Folner condition, i.e., for every positive number $\epsilon$ and every finite set $A$ of $G$ there exists a finite non-empty subset $F$ (Folner set) such that $|gF \Delta F|\leq \epsilon |F|$ for all $g\in A$, where $\Delta$ is the symmetric difference.
Recall that the class of elementary amenable groups is the smallest class of groups containing all abelian and finite groups and closed under taking subgroups, quotients, extensions and directed unions.
It is known that every elementary amenable group is amenable but the converse is false.
Is there an "strong form" of Folner condition which is equivalent to being elementary amenable for a given group? What about solvable groups?
By an "strong form" of Folner condition I am wishing some further conditions on the Folner set $F$ in the above, for example.
Another way, if we assume that the size of the Folner set (which is depending to $A$ and $\epsilon$) is bounded above by a function $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ of $\epsilon$ only. Under what conditions on $f$ one can conclude that the group is virtually solvable.
