Solution
= Solution
Write an element of $H\wr\mathbb Z$ as $(f,t^n)$. If it is central, commuting with $t$ makes the finitely supported lamp configuration $f$ invariant under translation. The only such configuration on the infinite set $\mathbb Z$ is the identity, so $f=1$. If $n\ne0$, then $t^n$ moves a nonidentity lamp at position zero to position $n$ and does not commute with it. Therefore $n=0$, and
$$
\boxed{Z(H\wr\mathbb Z)=1.}
$$