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 2generator amenable group? Perhaps easier, is it true that every countable amenable group embeds into a finitely generated amenable group?

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), 327352]). 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
$$
1\longrightarrow M \longrightarrow G \longrightarrow \mathbb Z \longrightarrow 1,
$$ 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 $K\le [H,H]$. 


By a result of B.H. Neumann and H. Neunmann (Neumann, B. H.; Neumann, Hanna Embedding theorems for groups. J. London Math. Soc. 34 1959 465479.), every countable solvable group of class $c$ embeds into a 2generated solvable group of class $c+2$. For finite groups one can use the following construction. Consider the group $S_\infty$ of finitary permutations of $\mathbb N$ (all permutations with finite support). It is generated by transpositions $(1,2), (2,3),...,(n,n+1),...$. The shift $n\mapsto n+1$ induces an injective endomorphism of $S_\infty$ into itself. Consider the (ascending) HNN extension of $S_\infty$ corresponding to this endomorphism. The resulting group is elementary amenable, 2generated, and contains all finite groups as subgroups. I do not know of any results about embeddings of countable nonelementary amenable groups into finitely generated ones. 


Theorem 2 of Hall, P. On the finiteness of certain soluble groups. Proc. London Math. Soc. (3) 9 1959 595622. shows that there is a finitely generated solvable group of derived length 3 with a subgroup isomorphic to ℚ The group is 3generated. I guess it shouldn't be too difficult to provide a 2generated example. A related question: is every countable elementary amenable group embeddable in a 2generated elementary amenable group? 


Yes, The Grigorchuk group embeds into a 2 generated amenable group which is also finitely presented. For a reference you can look at the paper of Grigorchuk titled "Solved and unsolved problems around one group". 

