All $L^2$-Betti numbers of a finitely generated group $G$ with an infinite amenable normal subgroup are 0, by a result of Gromov. (See Theorem 7.2 of Lück's book *$L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory*.) Such a group has deficiency $\leq1$, with equality only if 
$G\cong\mathbb{Z}$ or $c.d.G=2$. (See "$L^2$-homology and asphericity",
*Israel J. M.* 99 (1997), 271--283, by Hillman.)
Thus if $G$ has deficiency 2 its centre must be finite.