Let $G=\langle X;R\rangle$ be a finitely presented group. The rank of $G$ is defined to be the size of smallest generating set of $G$. The deficiency ${\rm def}(G)$ of $G$ is defined to be the maximum of $|X| - |R|$ over all finite presentations $G = \langle X;R \rangle$.

The deficiency of an amenable group can be at most $1$. One (maybe the only?) way to see this is to note that there is a Morse inequality for $\ell^2$-homology $$1-{\rm def}(G) \geq b_0^{(2)}(G) - b_1^{(2)}(G) + b_2^{(2)}(G),$$ where $b_i^{(2)}(G)$ denotes the $i$-th $\ell^2$-Betti number of $G$. Cheeger and Gromov showed that an amenable group satisfies $b_i^{(2)}(G)=0$ for $i \geq 1$. This implies in particular that ${\rm def}(G) \leq 1$ for $G$ amenable.

Now, apart from $\mathbb Z$ and Baumslag-Solitar groups $BS(1,n) = \langle a,b; a^nba^{-1}b^{-1} \rangle$, I do not know of any amenable groups which realize ${\rm def}(G) =1$. In particular, I do not know any examples of rank $\geq 3$.

Question:Does every amenable group of deficiency $1$ have rank $\leq 2$.

It can be shown that any amenable group with ${\rm def}(G)=1$ must have cohomological dimension $\leq 2$, which puts severe restrictions on $G$. What else is known about amenable groups with deficiency $1$?