This question arises from http://mathoverflow.net/questions/30653/hnn-embedding-theorem-for-amenable-groups/76261#76261
Recall that a group $G$ is called SQ-universal if every countable group is isomorphic to a subgroup of a quotient of $G$. The first non-trivial example of an SQ-universal group was provided by Higman, Neumann and Neumann in 1949. They proved that the free group of rank $2$ is SQ-universal, which is equivalent to the statement that every countable group embeds into a $2$-generated one. Presently many other examples of $SQ$-universal groups are known (e.g., hyperbolic and relatively hyperbolic groups).
It is straightforward to see that any SQ-universal group contains a non-abelian free subgroup and hence is non-amenable. However the following problem seems open.
Problem 1. Does there exist a finitely generated amenable group $A$ such that every countable amenable group embeds into a quotient of $A$?
I believe, the answer is "no". One way to disprove it would be to use the Folner functions, defined by Vershik in 70's. Recall that for a finitely generated amenable group $A$, $Fol_A\colon \mathbb N\to \mathbb N$ is defined to be by $Fol_A(n)$ = the size of a smallest finite subset $\subseteq S \subseteq A$ satisfying $|\partial S|/|S|\le 1/n$. The asymptotic growth of $Fol_A(n)$ is independent of the choice of a finite generating set of $A$ up to a natural equivalence.
It is not hard to show that, when we pass to subgroups and quotient groups, this function does not decrease in the sense of the natural relation
$$
f\preceq g \; {\rm iff}\; \exists\, C>0\; {\rm such\; that}\; f(n) \le Cg(Cn)\; \forall\, n.
$$
Thus to answer Problem 1 negatively it would be sufficient to prove the following.
Conjecture 2. For any function $f\colon \mathbb N\to \mathbb N$, there exists a finitely generated amenable group $A$ such that $f\preceq Fol_A$.
Erschler [On isoperimetric profiles of finitely generated groups, Geom. Dedicata 100 (2003), 157–171] showed the existence of amenable groups with $Fol$ growing faster than any iterated exponential function. She also announced the proof of Conjecture 2 there, but I did not find it in her later papers.
Final remark: Problem 1 also makes sense if we replace "finitely generated" with "countable".
