SQ-universality in the class of amenable groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T03:21:50Z http://mathoverflow.net/feeds/question/76286 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76286/sq-universality-in-the-class-of-amenable-groups SQ-universality in the class of amenable groups Denis Osin 2011-09-24T20:44:46Z 2011-09-25T02:19:30Z <p>This question arises from <a href="http://mathoverflow.net/questions/30653/hnn-embedding-theorem-for-amenable-groups/76261#76261" rel="nofollow">http://mathoverflow.net/questions/30653/hnn-embedding-theorem-for-amenable-groups/76261#76261</a></p> <p>Recall that a group $G$ is called <em>SQ-universal</em> 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).</p> <p>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.</p> <p><strong>Problem 1.</strong> <em>Does there exist a finitely generated amenable group $A$ such that every countable amenable group embeds into a quotient of $A$?</em></p> <p>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. </p> <p>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<br> $$f\preceq g \; {\rm iff}\; \exists\, C>0\; {\rm such\; that}\; f(n) \le Cg(Cn)\; \forall\, n.$$ Thus to answer Problem 1 negatively it would be sufficient to prove the following.</p> <p><strong>Conjecture 2.</strong> <em>For any function $f\colon \mathbb N\to \mathbb N$, there exists a finitely generated amenable group $A$ such that</em> $f\preceq Fol_A$.</p> <p>Erschler [On isoperimetric profiles of finitely generated groups, <em>Geom. Dedicata</em> <strong>100</strong> (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. </p> <p>Final remark: Problem 1 also makes sense if we replace "finitely generated" with "countable". </p> http://mathoverflow.net/questions/76286/sq-universality-in-the-class-of-amenable-groups/76299#76299 Answer by Mark Sapir for SQ-universality in the class of amenable groups Mark Sapir 2011-09-24T23:22:14Z 2011-09-25T02:19:30Z <p>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. </p> <p><b> Update </b> 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. </p> http://mathoverflow.net/questions/76286/sq-universality-in-the-class-of-amenable-groups/76305#76305 Answer by Simon Thomas for SQ-universality in the class of amenable groups Simon Thomas 2011-09-25T01:35:07Z 2011-09-25T01:35:07Z <p>Here's what you were looking for:</p> <p>MR2254627 (2007k:20086) Erschler, Anna(F-PARIS11-M) Piecewise automatic groups. (English summary) Duke Math. J. 134 (2006), no. 3, 591–613. 20F65 (20F69 43A07 57M07) </p> <p>"The main result of the paper under review is stated as follows: For any function f:N→N there exists a finitely generated group of an intermediate group (and thus amenable) whose Følner function satisfies FølG,S(n)≥f(n) for all sufficiently large n."</p>