# SQ-universality in the class of amenable groups

This question arises from HNN Embedding Theorem for Amenable Groups?

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 by $Fol_A(n)$ = the size of a smallest finite subset $S \subseteq A$ 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".

• An observation that might be stupid, but maybe is related to your queston: in G.N. Arzhantseva, V.S. Guba, L.Guyot, Growth rates of amenable groups, Journal of Group Theory, 8 (2005), no.3, 389-394 the authors proved that a f.g. amenable group can have balls that increase very rapidly. I am not sure it makes sense but maybe this behavior of the balls can be reflected in the fact that Folner's condition is verified for very large sets. Sep 24, 2011 at 21:15
• This is a nice observation, but unfortunately it does not help. The groups considered there are free solvable groups of increasing solvability degree. There are only countably many of them. Hence their restricted direct product (and therefore every such a group) embeds in a 2-generated amenable group. Sep 24, 2011 at 21:47
• In fact, the Folner function of every solvable group is bounded by an iterated exponential function, where the number of iterations is at most the solvability degree. It follows from the estimates in the paper of Erschler and the fact that every group of type $F/[R,R]$ embeds into the wreath product of $Z^n$ and $F/R$. In particular, all solvable groups have Fol bounded by some universal function. So there is no hope to prove Conjecture 1 just by using solvable groups. Interestingly, the solution announced by Anna uses locally finite-by-cyclic groups, which are elementary amenable Sep 24, 2011 at 22:21

Anna Erschler proved (the paper referred to in the question) that for every group $G$ with Foelner function $F$ the Foelner function of $G\wr G$ is $F^F$. This implies that if $G$ is SQ-universal in the class of amenable groups, its Foelner function $F$ must satisfy $F\equiv F^F$. I do not remember exactly but that probably means that one cannot prove the amenability of $G$ (at least existence of the Foelner sets) using Peano arithmetic. Perhaps a more detailed analysis of what happens to the Foelner function under taking subgroups and homomorphic images would immediately imply that $G\wr G$ cannot embed into a factor-group of a subgroup of $G$ if $G$ is amenable. You can also ask Anna directly.
Update Simon has answered the original question. But still I think it is interesting to find out if there exists a finitely generated amenable group $G$ such that $G\wr G$ embeds to a homomorphic image of $G$. If such a $G$ exists its Foelner function must be truly remarkable.