Does there exist an analog of the HNN Embedding Theorem for the class of countable amenable groups? In other words, is it true that every countable amenable group embeds into a 2-generator amenable group? Perhaps easier, is it true that every countable amenable group embeds into a finitely generated amenable group?



HNN Embedding Theorem for Amenable Groups?
If I am not mistaken, the answer is "yes".


**Theorem.** *Every countable amenable (respectively, elementary amenable) group embeds into a $2$-generated amenable (respectively, elementary amenable) group.*


The proof is based on the following lemma, which admits a quite elementary proof using wreath products (see [P. Hall, The Frattini subgroups of finitely generated groups, *Proc. London Math. Soc.* **11** (1961), 327-352]). Given a group $X$, we denote by $X^\omega$ the restricted direct product of countably many copies of $X$. 


**Lemma** (P. Hall). *Let $H$ be a countable group. Then there exists a short exact sequence
$$
1\longrightarrow M \longrightarrow G \longrightarrow \mathbb Z \longrightarrow 1,
$$  

where $G$ is $2$-generated and $[M,M]=[H,H]^\omega$.* 


In particular, the lemma implies the theorem when the countable group is perfect. To prove the theorem in the general case we use the following trick which goes back, I believe, to the paper by B. H. Neumann and H. Neumann cited by Mark.


Starting with a countable group $K$, consider the subgroup $H$ of the Cartesian (unrestricted) wreath product $K \, {\rm Wr}\, \mathbb Z$ generated by $\mathbb Z$ and the set of all elements of the base group $ f\_k\colon \mathbb Z\to K$, $k\in K$, such that $f\_k(n)=1$ for $n\le 0$ and $f\_k(n)=k$ for $n> 0$. Let $t$ be a generator of $\mathbb Z$. For definiteness let $t=1$. Then $t^{-1}f\_ktf\_k^{-1}$, considered as a function $\mathbb Z\to K$, takes only one nontrivial value $k$ (at $0$). This obviously gives an embedding $K\le [H,H].$


Moreover, it is easy to see that the intersection of $H$ with the base $B$ of the wreath product consists of functions $f\colon \mathbb Z\to K$ with the following property: There exists $N\_f\in \mathbb N $ such that $f(n)=1$ whenever $n\le -N\_f$ and $f(n)=f(N\_f)$ whenever $n\ge N\_f$. Obviously the map $\varepsilon\colon H\cap B\to K$, which maps every function $f\colon \mathbb Z\to K$ as above to $f(N\_f)$, is a homomorphism and $Ker\, \varepsilon$ is isomorphic to a subgroup of $K^\omega $. In particular, if $K$ is amenable (or elementary amenable), then so is $H\cap B$ and, consequently, so is $H$. Now applying the lemma to $H$ yields the theorem as $K\le [H,H]$.


