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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 L"uck'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.